I second the KT Guest House. For what you get the prices are great. There's a nice little pool that hardly anyone uses, and the rooms are large enough to unpack/pack and assemble/disassemble most bikes. Hallways and staircases are nice and wide, and there's a good sized garden/common area if you need to do any other work. Boxes can be stashed for free in the storage area beneath the steps. Also, no need to pay anything ahead of time if you book online or by phone.
As onrrbike noted, though, it is in a fairly quiet area, though it's not too far from the subway.
Lately, I've also been using Shanti Lodge in the Dusit District off of Samsen, north of Banglamphu. They're bike friendly, will keep boxes, etc., but you probably won't be able to store your bike in your room. You can lock it up on the main floor, though. They always find room for mine, and always out of the way.
KT is a bit better in terms of being bike-friendly, but I think that Shanti is in a much better area for food, riding, etc.
One other good thing about Shanti is that it is close to Velo Thailand, a pretty decent bike shop off of Samsen a few soi's up from the Banglamphu Canal. Just in case you need some parts, service, or if you want someone else to package up your bike for the trip home.