Here's an idea from BaB contributor Sports2 in the comments:
Rip is basically expiring. The only taker I can think of for him is a team that sucks and will be under the cap. Send him to New Jersey or Charlotte for a sack of doorknobs. This gives us enough money to re-sign Ronnie Brewer, who played much better and much more consistently.
And don't forget: younger and likely healthier.
I think this makes sense. Rip Hamilton was a win-now acquisition and the Bulls didn't win. He was an upgrade over Keith Bogans. He sometimes looked very good, especially during the rare time he played with Derrick Rose, including 19 points in 26 minutes in game one of the playoffs.
But his season was an overall disappointment, least of which was missing 38 games due to groin and shoulder issues. He showed good playmaking ability but his TS% sunk to a career-low 50% while still aggressively pulling the trigger. The real obvious signs of decline looked to be on the defensive end. He may have been a bit of a turd in the locker room and not forthright about his injuries. All told, by the end of the season Rip had shown to be better than Keith Bogans but stayed in the Bogans role. Both Kyle Korver and Brewer took the 'closing' SG spot at times showing that Rip hadn't 'won' the job according to Thibs.
Without Rose and Deng to start the year, going with another defensively-emphasized player like Brewer over Rip (plus Jimmy Butler taking some SF minutes) may be trouble. But if Rip's only really effective playing off of Derrick Rose, he won't be of much use this season, and there's little reason to believe he'll be healthier (or better) as he turns 35 in the middle of next season. The Bulls will need bodies to take up minutes and Brewer's proven to be better at that.
So if it's possible that a team under the cap (or with a big-enough trade exception...Utah?) would take Hamilton and his $5m (plus $1m buyout next season) off of the Bulls hands for effectively nothing, would you rather $4.37m instead go to Brewer?
Keep in mind, this would be the kind of move lauded as 'shrewd' while it's really the Bulls dodging the luxury tax. Kind of like signing Rip in the first place because he was willing to take a 2-year deal. But these are the kind of choices that they force themselves in.