I read an article recently about a proposed warp drive. The idea is that if space/time can be distorted in front and behind a vessel, it could be propelled faster than the speed of light. This is a practical but boring solution. Douglas Adams came up with the Infinite Improbability Drive (change the laws of probability so that something as improbable as faster-than-light travel can be probable,) and Harry Harrison had his bloater drive (stretch the ship like a rubber band, and when the nose has reached its destination, relax it forward in the same direction.) I've had some ideas in the same vein:
The Perception Drive: This one is the easiest: convince the astronauts that they have achieved light speed. If they're sufficiently sure of it, then objective reality won't matter.
The Escort Drive: Einstein once noted that a second on a hot stove feels like an eternity, while an hour with a beautiful woman can go by in a second. Therefore, if the ship is co-habitated by your dream woman (or man, if that's your thing) relative time will pass instantaneously.
The Alternate Universe Burn: The problem with faster than light travel is that the faster you go, the more your mass increases. Therefore, more and more energy is required to accelerate. Pushing an object past light speed would require more energy than is available in the entire universe. However - if, as quantum physics suggests, we live in a multiverse of uncollapsed possibilities, those infinities of universes with alternate outcomes could be burned as fuel, making faster than light travel possible. The downside is that there's always the possibility of astronauts in another reality burning up this one.