I came up with a few things that could be what a modern day GTA 3 is should a remake or sequel be made starring Claude. Fell free to discuss and add stuff that you find relevant.
1) More detailed Liberty City but retaining the dark atmosphere by keeping daytime activities to minimum and have the city more vibrant at night.
2) Drug dealers, prostitutes and what makes Red Light District come to life.
3) Luigi's Club, Marco's Bistro, Supa Save! and all other notable places enterable and interactive
4) Much more varied weapons and upgrades
5) Having Claude retain his mysterious personality and speaking a few words when needed such as Adam Jenson in Deus Ex: HR. A completely mute character wouldn't go so well in a modern game.
6) Reworked physics, advanced car handling and swimming
7) Bikes and helicopters
8) Varied missions