Was the nasi position hereditary?

No less than three different lists of nesiim appear in the book of Bemidbar.

In Bemidbar 2-7, the twelve tribal leaders are listed.

In Bemidbar 13, the twelve spies are listed, and described as nesiim. However, it would not make sense to send the tribal leader on an unofficial mission to hostile territory, so we may presume these were prominent people, but not the official leaders of each tribe. Indeed, the names all differ from Bemidbar 2-7.

In Bemidbar 34, the twelve tribal leaders are listed again. Once again, these names differ from the previous lists (Kalev ben Yefuneh was a spy and now is a leader; I have not checked closely, but I think he's the only overlap).

It makes sense that the names in Bemidbar 34 differ from Bemidbar 13, since the spies except Yehoshua and Kalev all died. It also makes sense that the names in Bemidbar 34 differ from Bemidbar 2-7, since 38 years have passed and the previous generation of leaders have all died.

What is noteworthy, though, is that the fathers of each leader in Bemidbar 34 are not the same as the names in Bemidbar 2-7. That suggests that the position of tribal nasi was not hereditary at this time.