As we know, Counselor Troi's mother is Lwaxana TroiLwaxana Troi and her father was Ian Andrew TroiIan Andrew Troi who died suddenly in 2343.
How are they all named Troi?
The most likely to me seems to be that Ian took Lwaxana's name based on the following evidence:
Lwaxana is very insistent adhering to Betazoid tradition (e.g. insists on being naked during wedding ceremonies)
Betazed appears to be matriarchal and matrilineal
Lwaxana did not take any of her other husbands' names
From these point, I'm guessing Troi was not Ian's unmarried name.
But this is largely circumstantial. Is there evidence that this line of reasoning is correct? Did Ian take Lwaxana's name? Alternatively, is there some evidence that Lwaxana took Ian's name?