you can look at the deadline trades of every cup winner and see the players who just missed out
Usually contending teams aren't dealing veterans. OP was ideally looking for guys who were with a team for a lengthy period of time. But for funsies:
1995: Devils traded away Corey Millen (50 points the previous year for NJ), Slava Fetisov (he'd get two Cups eventually with Detroit), and Alexander Semak (couple years removed from a 79 point season)
1996: Avs dealt Owen Nolan early in the season to get a PPQB in Sandis Ozolinsh. Jocelyn Thibault, Martin Rucinsky, and Andrei Kovalenko were away to get Patrick Roy and Mike Keane.
1997: Wings didn't resign Dino Ciccarelli in summer 1996. Keith Primeau was a contract holdout and traded with Paul Coffey to get Brendan Shanahan.
1998: ?
1999: Greg Adams spent the previous 3+ seasons in Dallas but wasn't resigned in summer 1998.
2000: Brendan Morrison was a contract holdout. He'd eventually sign but requested a trade. Denis Pederson was an alternate captain on that squad but wasn't happy about his diminishing role and being forced to play RW instead of C. Lou would package both to get Alexander Mogilny. Lyle Odelein also requested a trade because of an ongoing feud with head coach Robbie Ftorek; Odelein had won a Cup previously with Montreal. But still an unusual circumstance for somebody to ask off a contending club.
2001: Aaron Miller had been in the Nords/Avs system since 1993. He didn't play enough games to get on the Cup in 1996. In 2001, he'd be included in the Rob Blake trade.
2002: ?
2003: Boring offseason/deadline for the Devils. Bobby Holik had left via free agency but was part of two other Cup teams. John Vanbiesbrouck was coaxed out of retirement at the 2002 deadline to join the Devils for the stretch run. But he retired again in summer 2002. Vanbiesbrouck missed the Rangers Cup by a year, joined the Devils for their Cup defense in 2001 that fell short, and then retired before the Devils 2003 Cup.
2004: The aforementioned Vinny Prospal left via free agency in summer 2003. I wouldn't have guessed that Prospal was their leading scorer in 2002-03. Other than replacing Prospal with Cory Stillman, they mostly kept the band together and added Darryl Sydor.