Sounds like what happened to my old mac clone from bad RAM. If you take your iBook apart, is there more than one RAM module in there? If so, try running with just one, then just the other--assuming you still meet the 128 MB OS X requirement. If that's not possible, maybe try RAM from another iBook?
I don't remember clearly, but I assume those iBooks are like my PowerBook G3: one RAM slot that's user-accessible, and another that you must take the thing apart more to get to.
Bad RAM is my guess. If not, then some other hardware failure in need of repair. It doesn't sound like software to me, or you ought to be able to at least boot to an installer CD.