So, I would assume the answer to this question is "No", but I can't actually verify it anywhere. The criteria for getting your name on the cup is to play at least 41 regular season games for the cup winning team. But if a player is traded after that 41 games, is his name still engraved on the Cup even though he played out the season with another team?
According to this article:
Prior to 1977 only players who had completed the Stanley Cup playoffs were eligible. Today, players appearing in 41 regular-season games or one Stanley Cup Final game for the championship team have their names engraved on the Cup. The NHL makes exceptions for players who do not meet the standard because of injury or other extenuating circumstances.
And according to the NHL:
To have one's name engraved on the Stanley Cup certain requirements must be met. A player must have at least 41 games played with the club or one game played in the Stanley Cup Finals. However, in 1994 a stipulation was added to allow a team to petition the Commissioner for permission to have players' names put on the Cup if extenuating circumstances prevented them from being available to play.
However, according to Wikipedia:
The Detroit Red Wings received special permission from the NHL to inscribe the name of Vladimir Konstantinov, whose career ended after a car accident on June 13, 1997, on the Stanley Cup after Detroit defended their title in 1998.
So it appears that player traded away could be added in theory, but probably not in practice.