Wiktionary defines je ne sais quoi as
An intangible quality that makes something distinctive or attractive.
She has a certain je ne sais quoi about her.
Is there a single-word equivalent?
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The one-word option that is the closest match to the French original is "something."
She has a certain something about her.
This has the same figurative meaning and close to the same literal meaning as the French phrase.
There's a Cole Porter song called "She's Got That Thing" that uses "thing" to express the same thing as "je ne sais quoi":
She's got that thing, that certain thing,
That thing that makes birds forget to sing
But in modern English, I think "something" is more idiomatic.
Je ne sais quoi is a loan phrase in English so it is English already. This phrase captures the idea more precisely than any other equivalent and has the sense of that indescribable elegance, so that would be why it is loaned from French.
One word equivalent would be something but it is used with the adjective certain and makes sense in a context.
An explanation from http://french.about.com:
Elle a un je-ne-sais-quoi qui m'intrigue.
She has a certain something that intrigues me, There's something intriguing about her.
English's borrowing of this expression is limited to the meaning in the final example: "a certain something" or "something inexplicable." The only adjective that can accompany je ne sais quoi in English is "certain":
She has a certain "je ne sais quoi" that intrigues me.
His house is very nice, but it lacks a certain "je ne sais quoi."
or you can simply say "There's something about her." which would be the most common way to say.
Other than that, there is another phrase that has a similar sense in it: defy description.
Be so unusual or remarkable as to be impossible to describe:
the sheer scale of the Requiem defies description
If we really want to go with one descriptive word, panache comes close which has this sense of flamboyance also and is usually used to define people or things with that certain style. The origin is French again and it is used figuratively.
Cambridge Dictionaries defines as:
a stylish , original , and very confident way of doing things that makes people admire you
Thefreedictionary defines as:
distinctive and stylish elegance
It is also used with the adjective certain and you can find a lot of contemporary usages. So if we apply to your example:
She has a certain panache.
In The Daily Show's America: The Book, je ne sais quoi is half jokingly, but not entirely inaccurately translated as "it", as in "some people got it". Perhaps not the best term for formal writing, but "it" may be the closest equivalent that is actually used in English speech.
Original Quote (on page 127):
The French call it "je ne sais quoi." Americans, who aren't pretentious snobs and don't need fancy words to make them feel all cool, call it "it"
Charisma, “personal charm or magnetism”, is another term to consider. Charisma is perhaps not undefinable, but often is recognized without being pinned down to any specific characteristic of a person.
X factor or X-factor
X factor means some advantageous quality that someone or something is attributed to have, a quality that is regarded as mysterious and difficult to describe or isolate. (source)
Apparently it was coined in 1934 (source), but the recent popularity of the eponymous television show might make this word less appropriate for your purposes.
I'd say the context of what you're responding to will shape the word you're looking for.
If you just can't put description to the way something feels or looks, you could use: Unexplainable, Unfathomable, or Undefinable. These help pass the point across that it is something, but words can't explain how it feels.
Alternatively, if you're just not aware of a situation at hand. You would say that you were oblivious to it. Lacking the knowledge of that topic entirely.
Unsure could work as well. I personally think it doesn't feel as appropriate though and doesn't emphasize enough the point that could be made with other words.
There are all sorts of possible replacements for "je ne sais quoi". To borrow from two of the more erudite English speakers in the United States*, instead of:
"She has a certain je ne sais quoi", you could simply say, "She's cool".
or instead of, "The Mona Lisa has a certain je ne sais quoi", you could simply say, "The Mona Lisa is cool". Possibly followed by "yeah, yeah", for emphasis.
Seriously, the word "cool" gives the idea that something is interesting or neat, without specifying what exactly. Just like "je ne sais quoi".
*Beavis and Butthead