I do think it's in everyone's best interest to meet a player they revere like Mantle or DiMaggio if the opportunity ever presents itself.
I never got either's signature at an event, but my personal favorite player has always been Don Mattingly. I've encountered Mattingly at Yankees spring training, Marlins spring training, and Dodgers spring training and with either team he was the exact same guy. Always such a nice presence for fans. Always took the time to sign for everyone once he finished up for the day.
It just makes you like your favorite player that much more when he's also a really nice guy. I'm not sure I would continue to collect Mattingly if he weren't.
And now a card from one of those spring trainings:
