Favorite and most convenient are pretty different criteria for me. The most convenient for me is certainly Lincoln, because it is en route to most hikes in the Whites (coming from Vermont), and has all the services I need coming to/from: gas, cheap food, better food, pizza (GH), ice cream, the Mountain Wanderer, grocery store. Gorham seems to be in the same league, but I just don't go through there nearly as often.
But as for Lincoln, the downtown touristy section is kitsch and unattractive, so I wouldn't want to live there if I was relocating to the Whites. None of the towns in the Whites distinguishes itself in terms of nightlife, etc., anyway, so if I was moving to the Whites, it would make sense to be centrally located, which would probably mean Twin Mountain/Carroll.
(Of course, school districts would be a consideration if kids were in the picture.)