My proposal: a usable tool for working with line bundles on toric varieties and their cohomology. There are some tools (Polymake, Latte) for working with polyhedra, but I haven't seen a library dedicated specifically to toric varieties.
For example, you could provide a GUI tool for working with toric surfaces, where you can e.g. blow a point up by a single click (as far as I remember, this corresponds to adding an edge), compute intersections of divisors and cohomology of line bundles.
Also: toric deformations and degenerations, action of the Frobenius morphism (there are two descriptions of Frobenius push-forwards of line bundles, due to Thomsen and Bondal), finding exceptional collections etc.
I think this could be really interesting and useful.
