S.,
As a builder I'll throw some ideas your way. First, does your basement get damp at all, not when its flooded obviously but just in general? If so, carpet can hold the moisture in and even mold eventually. Just something to think about. If it does you may want to put some of the cheap laminate in. Home Depot or Menards has some really cheap stuff that looks nice when its down and is even less the installed carpet to put in. Plus its water resistant and won't mold like carpet. Toss an area rug down where the kids play & you're all set.
If it doesn't get damp down there, your carpet options are endless. Again the hardware stores have pre padded carpet that is about 15 feet across (I think, lol, don't quote me on it exactly). You can get large pieces and put it down or have it installed, again for really cheap. Another nice option is indoor/outdoor type carpet. If the basement floods again, it can be picked up and dried off, and put back down. . . assuming the flood is not sewage water or something gross.
Carpet stores are always good to try, but sometimes take more leg work to find a really good price with a color you like. The hardware stores tend to have some very cheap options.
Good luck!