Limiting it to players who played more games with the given team than any other, otherwise I could have about 4-5 players for the whole of the list...
Anaheim Ducks: Steve Rucchin
Boston Bruins: Ray Bourque (duh!)
Buffalo Sabres: Dave Andreychuk
Calgary Flames: Theo Fleury
Hartford Whalers/Carolina Hurricanes: Randy Ladouceur (yeah, I don't know, either)
Chicago Blackhawks: Jeremy Roenick
Quebec Nordiques/Colorado Avalanche: Joe Sakic
Columbus Blue Jackets: Rostislav Klesla
Minnesota North Stars/Dallas Stars: Tyler Seguin
Detroit Red Wings: Steve Yzerman
Edmonton Oilers: Ryan Smyth
Florida Panthers: Stephen Weiss
Los Angeles Kings: Rob Blake
Minnesota Wild: Wes Walz
Montreal Canadiens: Saku Koivu
Nashville Predators: Shea Weber
Kansas City Scouts/Colorado Rockies/New Jersey Devils: Scott Stevens
New York Islanders: Pat LaFontaine
New York Rangers: Adam Graves
Ottawa Senators: Shawn McEachern
Philadelphia Flyers: Eric Desjardins
Winnipeg Jets/Phoenix/Arizona Coyotes: Teppo Numminen
Pittsburgh Penguins: Mario Lemieux
St. Louis Blues: Dallas Drake
San Jose Sharks: Owen Nolan
Tampa Bay Lightning: Fredrik Modin
Vancouver Canucks: Trevor Linden
Washington Capitals: Calle Johansson
Atlanta Thrashers/Winnipeg Jets: Yannick Tremblay
Vegas hasn't been in the league long enough for me to have a favourite yet.