How should I call the grandson of my aunt?
Should I call him cousin or something else? What is the generally accepted term?
There are many terms that could be used for this relation. Even though technically he is your second-nephew, but mostly the phrase "first cousin once removed" is used.
According to answers of a similar question on Quora:
The “technical” genealogical term is “first cousin once removed.” You and your cousin are of the same generation, sharing grandparents. Your cousin’s child is one generation younger than you, so “removed” one generation
Wikipedia also defines the mentioned term as "A first cousin's child"
That would be first cousin once removed, who is a first cousin's child.
Call him by his name. If his name is Davit, you call him "Davit".
You can say he is "a cousin". That is correct (it means a relative other than an ancestor, descendant, or brother/sister/aunt/uncle), and usually enough detail.
Last week, Davit, a cousin of mine, came to visit. He's a nice boy, in 9th grade and doing very well in maths.
If you need to be specific, you can describe him as "the grandson of my aunt" That is simple and clear and given a moment everybody will understand it.
Unless you are engaged in genealogy (the study of family trees) don't use "first cousin once removed". Most native speakers of English (including me) have only a vague idea what this means and you will need to explain. It is also ambiguous, as the nephew of your grandfather is also a "first cousin once removed".
You will be his second uncle. I guess your aunt is your mother's or father's sister. And your cousin had the baby. So you are his second uncle. You can't be his first uncle or uncle because you are not the brother or the sister of your aunt's children. You should call him second nephew . nephew or niece if she is a girl.