Both
Unreal Tournament and
Quake III Revolution preceded the release of the PS2 Network Adapter, but they would have most likely made for some good online twitch shooting action.
With the server shutdown of Twisted Metal: Black Online, it would have been nice if
Twisted Metal: Head On - Extra Twisted Edition had included an online multiplayer feature on the PS2.
RoadKill would have also made for a good online title, because the more online vehicular combat, the better.
Burnout 2: Point of Impact was a huge step forward for the Burnout series, and online multiplayer would have been the cherry on top of an already great game. Not to mention the fact that the Burnout series was published by Acclaim instead of EA Games at the time, so it's online servers could have very well survived the subsequent server shutdowns of it's successors. The futuristic racer
XGRA: Extreme-G Racing Association would have made for a unique online racing experience, with it's variety of game modes, weapons and tracks.
Despite their PS2 ports not being developed by Neversoft, both
Tony Hawk's Project 8 and
Tony Hawk's Proving Ground could have benefited greatly by including online multiplayer.
Probably one of most obvious choices,
TimeSplitters 2 had the potential to be one of the PS2's greatest online shooters, had Free Radical Design finished refining a proper online mode. Not to take anything away from the included LAN mode, of course. Similar case as with Burnout 2, since the TimeSplitters series was published by Eidos Interactive instead of EA Games at the time,
TimeSplitters 2's online mode could have very well survived the server shutdown of TimeSplitters: Future Perfect.