Mathematical results don’t have to come from a bolt of lightning inspiration. This is a list I made long ago which may not help anyone who’s trying to do research, but may help someone who isn’t understand a bit more about the process.

• do the homework: look at existing results and see if they can be improved.
• do the non-assigned problems: look at merely analogous existing results and see if they can be adapted.
• do the starred problems: look at open questions and see how they relate to yours.
• change your perspective: look at equivalent characterizations of the concepts you’re working with, and see if that buys you anything.
• stand on your head: try to prove the negation of the statement you’re working with and try to pinpoint why that fails. (Or maybe it won’t.)
• increase your options: look for new equivalent characterizations of the concepts you’re working with.
• widen your view: look for other related notions and compare them to yours in the way you are trying to compare yours to each other.
• move to a sub-setting: prove a special case or assume a helpful lemma.
• move to a super-setting: generalize your problem and try to prove that. This helps more often than you might expect.
• change the problem: modify the definitions to see if you can prove results for those changed concepts.
• olio: combine the above in every way you can think of.