I don't go to enough games and don't sit in the seats where it's likely I'll have a remote chance at a ball from a milestone.
But for me it would depend on the player. If it was a player I really liked I'd probably give it back especially if I could actually hand it to them instead of a team lackey.
Players I didn't like? Probably not.
Arod, Sheffield (Yes, I know he's retired) guys like that nope, unless they really made it worthwhile.
Jeter would have been a harder decision. I'm not really a fan of his, I always felt he got a lot of hype - not his fault, but still annoying. And was always just a bit too smooth and polished. But even being a Sox fan I have a lot of respect for how he played and some of his attitudes towards the game in general and the rivalry. A player who says he appreciates being booed in Fenway because that's the game and that the fans don't generally do that with the average players can't be all bad. (And now they're both retired - Nomar is still better
)
Which is all odd now that I'm typing it.
A player I like would get back an item that I'd probably keep otherwise.
The player I didn't like would get back an item I'd probably just sell. (With some regret, since it would be a really great item. )
Not exactly logical, but it'll have to do.
Steve B