House of Finwë
We have the House of Finwë, which is comprised of 6 generations of Elves after him.
Finwë > Fingolfin > Turgon > Idril > Eärendil > Elrond > Elladan/Elrohir
Other than the House of Finwë (Noldor Elves), there doesn't seem to be enough information on the other Elven family trees, most notably the "pure" Vanyar and Teleri.
House of Thingol
The 2nd longest Elven family tree would be: the House of Thingol. It comprises of 5 generations of Elves after Thingol (a Sindarin Elf).
Thingol > Luthien > Dior > Elwing > Elrond > Elladan/Elrohir
As a side note, both Finwë and Thingol were the "Elven-ambassadors" responsible for leading their people in(to) Valinor. None of the Elves who first awoke were ever named, save Imin, Tata and Enel, and presumably Cirdan. However, we don't even know if Imin, Tata, and Enel exist!1 Cirdan is noted to be the longest-lived Elf by the time of the Third Age, but he doesn't have any descendants. So it would seem that Finwë and Thingol have the oldest named family trees.
1 See the third bullet point in Jason Baker's excellent answer: 'They never existed in the first place.'