The NBA maintains more nicknames in moves than any pro sports league:
NBA:
Philadelphia Warriors --> San Francisco/Golden State Warriors
Milwaukee Hawks --> St. Louis Hawks --> Atlanta Hawks
Vancouver Grizzlies --> Memphis Grizzlies
Minneapolis Lakers --> Los Angeles Lakers
San Diego Clippers --> Los Angeles Clippers
San Diego Rockets --> Houston Rockets
New Orleans Jazz --> Utah Jazz
Kansas City Kings --> Sacramento Kings
Charlotte Hornets --> New Orleans Hornets (Changed to Pelicans a decade after the move following sale of the team)
Rochester Royals --> Cincinnati Royals (Later changed to Kings only after moving to KC because of the MLB KC Royals)
(There is also an argument for Tri-Cities Blackhawks --> Milwaukee Hawks)
NFL:
Cleveland Rams -- Los Angeles Rams --> St. Louis Rams
Chicago Cardinals --> St. Louis Cardinals --> Phoenix/Arizona Cardinals
Oakland Raiders --> Los Angeles Raiders --> Back Again
Los Angeles Chargers --> San Diego Chargers
Baltimore Colts --> Indianapolis Colts
Boston Redskins --> Washington Redskins
(Although the Oilers retained their nickname through the first two seasons in Tennessee, the intent was always to change their name once their new stadium was open)
MLB:
Philadelphia Athletics --> Kansas City Athletics --> Oakland Athletics
Boston Braves --> Milwaukee Braves --> Atlanta Braves
Brooklyn Dodgers --> Los Angeles Dodgers
New York Giants --> San Francisco Giants
NHL:
Atlanta Flames --> Calgary Flames
(There is also an argument for Minnesota North Stars --> Dallas Stars)
MLS:
None
In fact, the NBA is the only major sports league since 2000 (Grizzlies and Hornets pre-sale) to move a team and maintain it's old nickname. We're actually kind of lucky we kept the SuperSonics name in Seattle.