I think you have to split the nod here between the Landeta and Collins. Landeta punted for 22 years in the NFL. He was the Giants' punter from 1985-1992, and half of 1993. Landeta was twice voted to the Pro Bowl as a Giant and once named All Pro.
Oddly, as good as he was, a glance through the Giants' media guide shows that Landeta holds no significant Giants' punting records.
Collins, of course, led the Giants to the 2000 Super Bowl. We don't want to remember the game itself, but he did get the team there. And he quarterbacked the team from 1999-2003. Giants' fans probably had a love-hate relationship with the sometimes erratic Collins, but there was a lot more good than bad.