I'm putting together a summer music program and (hurrah, hurrah) I got a bass clarinetist. However, music for bass clarinet is limited. Very. So my question is: can bass clarinet play music meant for bassoon? Or there another instrument's music that bass clarinet can use? Please help me.

If the bass is a low C one the bottom note is the same as a bassoon - you just miss the bottom 3 notes if it's an Eb one. The music itself should work reasonably well. The main problem can be reading the music. Hopefully the player will be able to sort out reading bass clef & transposing up a tone at sight. If not it will mean a lot of work transposing the parts up into treble clef & writing them out or using Sibelius (or other software).

Yes it can. Bass clarinets often double bassoons in orchestras/wind bands, and an advanced model (as previously pointed out) can achieve all the notes a bassoon has. Chances are a student won't have a lower model, but those three missing notes won't be a huge problem - it can go up an octave easily.
Regarding transposition, a bass clarinettist should be able to read bass clef, even if not particularly well, and as a clarient player at all should be able to transpose well. Otherwise it shouldn't take too long to transpose a couple of parts if you have the right software - I've done it for bari sax-bass clarinet for a jazz band a couple of times.
I've bought a few bassoon and cello peices to play on my bass, and they're all fine. The biggest problem you might have is if they have to go above about F4 (at concert pitch), or the G at the top of the stave in their own music, or F with 2 legers in bassoon music - this is where they will have most trouble sounding notes. I doubt you'll need them to go that high, though!

If your bass clarinetist is not particullarly good at transposing or reading bass clef...tenor sax parts are in treble clef and in Bb just like the bass clarinet...that might work well for younger students...or Baritone/Euphonium Treble Cleff parts...

If you play anything other than bass clarinet music on a bass clarinet then it is no longer music and hence you cannot play anything else on it. If you are disapointed because there is no music then its tough luck. I guess proper composers who write proper music don't think very much of bass clarinets.