A reasonable aim of formal mathematics/logic is to build systems which can "interpret" many things. As an example, ZFC can interpret a number of things. Incompleteness Theorems provide us with a conclusion that there are inherent limitations in formal systems. Turing's doctoral thesis (Systems of Logic based on Ordinals) explored this idea where we can overcome this inherent incompleteness by appending these unprovable formulae to existing system to arrive at a more complete system.
But it appears to me that there is nothing much remarkable in employing this strategy for arriving at more general system/stronger theory. Ideally, we should be making system much more general not by adding stuff piece by piece, but by "expanding" the system more generally. I read somewhere Godel suggested looking for stronger axioms of infinity.
My question is the following: how do mathematicians/logicians devise more general systems? What kind of things do they keep in mind when they step out to expand their systems? What are the ground rules to build general systems (or stronger axioms?) which can interpret my current system (and much more)?