NBA:
Utah Jazz doesn't really make much sense in a town like Salt Lake City. So how about New Orleans Jazz gets re-founded in their original city, with the removed "Jazz" being named something like Utah Mormons, or, to make it religiously neutral, Utah Saints? ;-)
NHL:
Colorado Avalanche could keep their (Québec) "Nordiques" sound by making it something like Colorado Nordics.
German Bundesliga (football):
When people are having the debate of tradition vs. commercialisation, I always point out that the supposedly traditional club 1. FC Köln (Cologne) was in fact a merger. So maybe they could keep some of their older tradition by keeping the older name "Kölner Ballspiel-Club 1901", shortly Kölner BC 1901.