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.
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.