Look at flabbier objects. This seems to be especially useful in complex algebraic geometry. Hard to prove something for varieties? See if there's a version that's true for schemes. Or maybe Kahler manifolds. Or worse: stacks. Vector bundles giving you trouble? Try coherent sheaves. Try quasi-coherent sheaves. In fact, try complexes of them. This is really just a special case of "Generalize the question as far as you can" but in this specific case, it's rather clarifying, here are some examples in algebraic geometry:
- It's hard to say anything about fundamental groups of complex projective varieties that isn't also true about compact Kahler manifolds. Perhaps the proof should focus on using the Kahler structure, when you're working on these.
- Want to parameterize subvarieties of a projective variety? Tough, it doesn't work. SubSCHEMES, however, gives the Hilbert Scheme.
- Proving things about ideals is often easier to do with modules in general