I think two general tendencies can be found here, one being "Let's make a free version of <game X>!" and the other one "Let's make a game that is like <X> but with such and such features added (or improved)". There are of course cases where this tendencies are more or less clearly present, while sometimes the developers do not openly proclaim their source of inspiration at all, leaving it to the players to decide.
Below is a list of such games that I've found so far. I've tried not to include projects that make use of copyrighted material (e.g. resources from the original game), and engine recreations that require original game files to play are also out.
1. Free content replacement for original engine/engine recreation
Battle Isle 2200 Doom
- Freedoom (Doom engine)
- Blasphemer (Doom engine)
- Zauberer (Doom engine)
- Open Quartz (QuakeForge, DarkPlaces)
- OpenQuartz 2 (DarkPlaces)
- OQPlus (QuakeForge, DarkPlaces) [SVN]
A-Train Carrier Command Chip's Challenge Crayon Physics Crimsonland Dungeon Keeper Elasto Mania Elite Escape Velocity Guitar Hero Herzog Zwei Lemmings Lode Runner Master of Magic Microsoft Flight Simulator Minecraft OutRun Paratrooper Planet Defender Railroad Tycoon Rescue! RPGMaker Robot Odyssey Scorched Earth Secret Agent Sentinel The Settlers II: Veni, Vidi, Vici Sid Meier's Civilization II Sid Meier's Colonization SimCity Stars! Stunts Super Mario Bros. Super Mario Kart Total Annihilation Transport Tycoon Deluxe Turrican Warlords Warlords II Wing Commander Wing Commander: Privateer Worms 3. Free games inspired by classic titles (not exact remakes)
Age of Empires Anno series Battle Isle 2200 Battlefield series, Call of Duty series Bos Wars, Command & Conquer: Red Alert BoulderDash, Emerald Mine, Supaplex, Sokoban Castle of the Winds, Ancient Domains of Mystery Castlevania series Celtic Legends, Heroes of Might and Magic, Civilization Command & Conquer series Dark Colony Descent Diablo Doom Doom, Quake etc. Dune II Earthworm Jim Empire Deluxe Fleet Command, Harpoon Frontier: Elite II Gradius, Castlevania, R-Type Grand Prix Legends Grand Theft Auto series Gun Half-Life: Counterstrike Heroes of Might and Magic series KKnD, StarCraft Lost Vikings Master of Magic Master of Orion Natural Selection Oxyd, Rock'n'Roll Populous Puzzle Bobble Quake III: Arena, Unreal Tournament R-Type Scorched Earth Sid Meier's Civilization II SimCity Star Wars: TIE Fighter, Descent: Freespace StarCraft Solar Fox Super Monkey Ball Turrican series Ultima series Warcraft II: Tides of Darkness Warcraft III: Reign of Chaos Warsong, Master of Monsters Warlords II Wizardry VII X-COM: Enemy Unknown 4. Board game implementations
Diplomacy Risk Space Hulk Titan