What is the appropriate word for husband of the daughter of my brother?
That is, son-in-law of one's brother.
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While nephew-in-law may be used, it is uncommon to use the "-in-law" epithet for anyone outside of an immediate family relation (that is: parents, children and siblings). It is more common to use a more explicit term, such as my niece's husband, possibly because nephew-in-law could be ambiguous - is it your niece's husband, or your spouse's nephew? Some people will also just call them their nephew.
If you wish to be specific that it is your brother's son-in-law, not your sister's son-in-law, there is no way of stating that in English other than how I just stated it.
As others have already stated, the "husband of the daughter of [your] brother" is your nephew-in-law. And as @Mitch noted there is a "Paucity of words for relationships in English" -- see Paucity of words for relationships.
The paucity of words in English for familial relationships can be regarded as either a weakness -- because imprecision can lead to confusion -- or a strength, because it allows for delicious nuances free of the danger of slander suits.
If you are fond of your nephew-in-law, you can call him simply "nephew"; you are implying he is as close as a blood relation. If you are neutral, call him your nephew-in-law. If you want to telegraph negativity, call him "my brother's daughter's husband". This accomplishes the same job as calling him "my niece's (pejorative word) husband" without being so impolite as to use a pejorative word.
The more distant the connection, the better this works. For example, no one would think I had a high opinion of "my husband's cousin's son's wife", whereas if I said "my cousin" [which is technically defensible]*, the hearer would assume a warm relationship.
- A relative by blood or marriage; a kinsman or kinswoman
Where the original question is:
"What is the appropriate word for husband of the daughter of my brother?"*
*(Note the title specified English; presumably the language)
I believe the other answers have provided the (unfortunate) answer you were seeking... assuming you were looking for a familial title that conveyed the mechanics of the relation viz. Daughter, Step-Father, Third Cousin twice removed, etc.
I only add my answer to suggest that the characterization of this fact as a "paucity" of the English language is perhaps misleading.
I would like to assert that the premise of the question is perhaps constraining a good deal of the flexibility of the language. I think that royal blood-lines probably serve as a useful metric in this regard, as such relationships and their proper nomenclature play an outsized role compared to non-royal familial relationships. To put it simply, if you want to address the husband of the 2nd-in-line-to-the-throne (of course you are King/Queen and we are referring to your firstborn daughter's husband assuming we don't have additional constraints such as male primogeniture), you simply create a Title (proper noun) for which this person is to be known and addressed. (see 'Philip, Duke of Edinburgh, a former prince of Greece and Denmark') However English that all may be, I do think it obscures the proper function and use of language; to transmit information from sender to receiver. Thus in conclusion, I might suggest that the answer for your question is not so much what would another person call the husband of their brother's daughter, rather it is what relationship do you want to convey as the speaker when communicating that relationship to some listener(s)? Perhaps "Friend", "Scoundrel", or just "The man who makes my niece very happy" would have more utility than a proper title or description of familial relationship. P.S. if in doubt, it’s hard to go wrong with "Mr."
Please go easy on me, it's my first answer :)