The Wikipedia article on Niger and the online Oxford Learner’s Dictionaries say that the proper term is Nigerien, as Vogel612 points out below. But The New York Times Manual of Style and Usage (1999) and Merriam-Webster’s Eleventh Collegiate Dictionary (2003) say that the proper name is Nigerois. And Merriam-Webster online now lists both Nigerien and Nigerois as correct forms. Is this simply a terminological difference between U.S. (or North American) English and British English, or is one word truly more appropriate than the other?
When I find multiple conflicting usages, I like to see what other people are using. The NGrams data clearly shows a vast preference for Nigerien. A slate.com article which claims to have verified its facts with the Nigerien Embassay in the US says
What do you call someone who hails from Niger? Old-schoolers (and, in what an editor there called "something of an oversight," the Merriam-Webster Online Dictionary) still use the archaic "Nigerois" (nee-zher-WAH); more common and up-to-date is "Nigerien" (nee-ZHER-yen).
An NYT columnist who wrote an article about Nigeriens, was told by his editor that the NYT style manual prefers Nigerois to avoid confusion with Nigerians. But while in Niger he was told that nobody uses that word and so decided to stick with Nigeriens.
So stick with Nigerien, unless you are happy to sound archaic for the purpose of preventing confusion.
If you wish to break it down by US/BR differences, you can:
As the question notes, different references have different opinions about the merits of Nigerien versus Nigerois. As Vogel 612 observes, both Wikipedia and the online Oxford Learner's Dictionary endorse Nigerien. In addition, the World Fact Book (1997), the New Oxford American Dictionary (2001), the New Oxford American Dictionary and Thesaurus with Language Guide (2003) offer only Nigerien as the word for the people of Niger. The Concise Oxford English Dictionary, Revised Tenth Edition (2001), is noncommittal: It includes an entry for the noun Nigerian ("a native or inhabitant of Nigeria") doesn't offer any advice on what to call "a native or inhabitant of Niger."
On the other hand The New York Times Manual of Style and Usage (1999) has this:
Nigerois (sing. and pl.) The people of Niger, The adjective is Niger. Do not confuse with Nigeria and Nigerians, for the larger country and its people.)
The U.S. Government Printing Office Style Manual (1984)—which gives "Nigerois (singular, plural)," though you may not be able to see anything in the online snippet view of this book—Robert Harris & American Heritage Dictionary, Secretary's Handbook (1984), Robert Grover, U.S. News & World Report Guidelines for Writers and Editors (1994), Amy Einsohn, The Copyeditor's Handbook: A Guide for Book Publishing and Corporate Communications (2000), and Patti Tasko, The Canadian Press Stylebook: A Guide for Writing and Editing (2006) likewise give only Nigerois as the name to use for the people of Niger. To my surprise, the Associated Press Stylebook (2002) doesn't offer any guidance on what to call the people of different nations.
Merriam-Webster's Eleventh Collegiate Dictionary (2003) lists only Nigerois, but the online version of MW's entry for Niger broadens the options to Nigerien and Nigerois:
— Nigerien adjective or noun
— Nigerois noun
Interestingly, MW online doesn't offer Niger as an adjective form of the country Niger, whereas the New York Times instructs its writers to use only that form as an adjective.
Arguments for 'Nigerien'
I am aware of four arguments that others have made for preferring Nigerien to Nigerois. First and most impressively (see the Ngram charts posted by Mr. Shiny and New 安宇 in his answer), Nigerien dominates current usage.
Second, as Slate Magazine's The Explainer (2004) argues in a commentary on Niger, the term Nigerois is "archaic":
Until 1960, Niger was a colony of France, and French is still Niger' official language. Its neighbor Nigeria, with which it is occasionally confused, was under English control before it gained its independence, also in 1960; that's why the pronunciations differ so sharply. What do you call someone who hails from Niger? Old-schoolers (and, in what an editor there called “something of an oversight,” the Merriam-Webster Online Dictionary) still use the archaic Nigerois (nee-zher-WAH); more common and up-to-date is Nigerien (nee-ZHER-yen).
Third, according to a fairly detailed Wikipedia user page posted by User:A12n, Nigerois is a made-up term with no historical legitimacy other than the fact that some people have used it:
Nigerois is a coined term of French construction that is not actually a word in the French language, but has had some use in English to refer to people from Niger, West Africa.
And fourth, referring to an October 11, 2005, "On the Ground" blog item by New York Times editorial columnist Nicholas Kristof—again cited in Mr. Shiny and New 安宇's answer—it may be argued that since "the people here [in Niger]" don't use the term Nigerois, English speakers not in Niger shouldn't use it either.
Arguments for 'Nigerois'
The strongest argument in favor of Nigerois—and undoubtedly the reason that U.S. periodicals like U.S. News & World Report and the New York Times adopted the term—is that it puts maximum distance between the names for the people of Niger and the people of Nigeria. Dixie Wills, New World Order: Every Country on Earth, Sorted (2008) puts the case (rather obnoxiously) this way:
Pub fact: For decades, the pubs, bars and discotheques of the world have resounded to the debate, often quite heated, surrounding the adjectival form of Niger. Go for 'Nigerian' and everyone thinks you're referring to Nigeria; opt for 'Nigerois' and no one knows what you're talking about; while plumping for plain old 'Niger' just doesn't sound right. Apparently, the answer is to aim at a noise approximating to 'Nigérien'. Good luck.
Obnoxious or not, Wills did enough research to discover that the River Niger owes it name to the Tuareg word n'eghirren (which she says means "flowing water"). On the other hand, she failed to notice that, virtually unanimously, the institutional proponents of Nigerois recommend using it only as a noun referring to the people of Niger, not as an adjective.
To (some) North Americans, the pattern of the word Nigerois is immediately recognizable as matching that of Quebecois (a native of Quebec), so it isn't an altogether outlandish formulation. But its difference from the noun Nigerian is the term's preeminent virtue.
The other arguments on behalf of Nigerois are perhaps more accurately characterized as attempts to counter the arguments in favor of Nigerien.
Regarding the massive advantage that Nigerien enjoys in popular usage, the counterargument is that Nigerois is predominantly a noun, whereas Nigerien piles up the vast majority of its matches in a Google Books search as an adjective. Since the Ngram charts that show the overwhelming relative frequency of Nigerien don't isolate occurrences of that word as a noun, they aren't terribly relevant to the question of how Nigerois as a noun and Nigerien as a noun match up.
As for the assertion that Nigerois is archaic, it's worth noting that the majority of the 97 Google Books search matches for Nigerois are from the the period between 1990 and 2014. The totals are 34 from the 29-year period of 1961–1989, and 63 from the 25-year period of 1990–2014. That hardly suggests the timeline distribution of an archaic term. The oldest instance is exceedingly old—from U.S. Department of Defense, Africa Names and Concepts (1961)—but only a handful of others come from the 1960s.
It's also worth pointing out that the sentence following the one from the Slate Explainer article that Mr. Shiny and New 安宇's answer quotes says this:
Still both [Nigerien and Nigerois] are acceptable, as long as you don't call your friend in Niamey a Nigerian.
Neither the criticism of Nigerois based on its etymological origin nor the one based on its nonuse in Niger seems especially compelling—unless we are determined to adopt a comparable test for the names we use in English for other nationalities of the world.
A quick look at 'Nigeran'
The Ngram charts posted by Mr. Shiny and New 安宇 that track the word Nigeran are intriguing, but unfortunately they, too, run afoul of a confounding variable—in this case, the water-soluble polymer (or fungal wall glucan) nigeran. Of the 27 matches from the period 1985–1987 in a Google Books search for Nigeran, 18 were to the polymer, and 9 were (arguably) to the adjective form of Niger. I say "arguably" because three of the latter group appear in publications titled Nigerian Economy: A Textbook of Applied Economics (1986), Equity and Trusts in Nigeria (1986), and Nigerian Journal of Palms and Oil Seeds (1987)—and in each case, the book contained only one match for Nigeran, suggesting that these instances might have involved typos for Nigerian. The remaining six instances were far from obviously Niger-related either; but when you're dealing with snippet views, the necessary context can be hard to come by.
Almost certainly the reason Nigeran didn't catch on as the adjective form of Niger is that typos and typesetting mistakes of this very kind were a constant danger. Several of the matches from the 1960s and 1970s clearly use Nigeran as an adjective for Niger; but in the 100 matches for Nigeran for the period 1972–1993, I couldn't find a single instance where Niger was unmistakably intended. Most matches were for the chemical nigeran, and many of the rest were clearly typos for Nigerian.
One nice thing about this search data for Nigeran is that it shows what a term that truly is archaic looks like in Google Books search results: Almost all of the matches turn out to be mismatches of one kind or another.
Ultimately, even though I don't find any of the four asserted arguments on behalf of Nigerien particularly persuasive, I would probably use it in preference to Nigerois, for two reasons. First, I am impressed that Merriam-Webster's online has switched from "Nigerois only" to "Nigerois and Nigerien" in the past decade. I don't believe that the Google Books evidence supports a claim that Nigerois is obsolescent or archaic, but I do think that Nigerien as a noun is gaining ground—and as an adjective it clearly enjoys a massive advantage, since the advocates of Nigerois generally don't recommend using it as an adjective at all.
Second—and more compelling, to my mind—I think that the rationale for avoiding Nigerien wasn't simply that the word looks a lot like Nigerian. It was also, to some extent, the fact that publishers didn't trust their audience to know that any such place as Niger existed. If you see Nigerien and you know only Nigeria as a country with a relevant-looking name, you're likely to suppose that Nigerien is a form of Nigeria. But thanks to the U.S. government's yellow-cake uranium fiasco of the mid-2000s and the attention now being paid to that area of West Africa and the area just south of it, many more English-speaking readers today are aware that Niger exists; and as a result fewer people by default associate Nigerien with Nigeria than was once the case. Niger as an adjective may still be common, but (in a situation mirroring Nigerien as a noun versus Nigerien as an adjective) those instances tend to get lost in the forest of instances of Niger as a noun.
The more frequently people use and recognize Nigerien as a noun or as an adjective associated with Niger and not Nigeria, however, the less reason anyone has to adhere to Nigerois. That reality doesn't make the latter term a living fossil, but it does suggest that the historical direction of usage is against it. The only complication hindering Nigerien in its progress toward universal adoption in English is its susceptibility to hard-to-catch typographical error—a practical problem that may keep Nigerois in the New York Times Manual of Style and other style guides for years to come.