Special Agent - a classic platform game inspired by Apogee's Secret Agent. Thanks to Scpm of 3D Realms Forums for mentioning this one! ^_^

Wolfram is a project to faithfully recreate Wolfenstein 3-D with modern graphics. Take a look at the screenshots here.

Renegade X is a creative re-imagining of Westwood Studios' FPS in the C&C universe, Command & Conquer: Renegade, made on the Unreal 3 engine.

Neil79 has posted a link at Reloaded.org Forums to a list of indie retro games at Overclockers UK, which includes many a remake of some oldies (Amiga & PC).

Chaos Reborn is a remake of the old classic game Chaos by Julian Gollop.

Here's a rather lengthy thread in a German forum that lists various classic game remakes (some links might be dead/broken though):
http://www.forum-3dcenter.org/vbulle...d.php?t=289931