You could ask them to discover (in effect) the spectral sequence for a specific composition of Exts and Tors, starting with a case where they can show that (say) Tor_n vanishes for n>1, and then proceeding to a case where Tor_n vanishes for n>2, (and so on for as far as you like) pointing them toward various interlocking exact sequences and prompting them to figure out how to organize all that information.

Then you could actually have them compute something specific using the general machinery they've developed.

An advantage of this project is that different students can be assigned different compositions of functors, giving problems that are different enough to discourage collaboration but similar enough to be graded on the same scale.