A very interesting heuristic is a principle in complex variables called "Bloch's heuristic principle".
Bloch's principle is about families of analytic functions called "normal". A family F of analytic functions on a domain D is called normal on D if every sequence of functions of F has a subsequence that converges uniformly on compact subsets (either to an analytic function or to infinity). Normal families are very studied for their applications in complex dynamics.
Bloch's principle goes as follows :
A family of analytic functions on a domain D having a property in common is most likely to be normal if there is no non-constant entire function having this property on the whole complex plane.
There are many examples of Bloch's principle. For example, take the property of being bounded : a well known theorem of Montel says that a family of analytic functions on a domain D which is uniformly bounded is necessarily normal on D, and Liouville's theorem says that there is no non-constant entire bounded function.
Or, take the property of omitting two distinct complex values. Again, a theorem of Montel says that a family of analytic functions on a domain D such that each function omits a,b in C, a different than b, is normal on D. The version for the whole complex plane is a well known theorem of Picard, that says that there is no non-constant entire function that omits two distinct complex values.
However, there are many counter-examples to Bloch's principle as it is stated, but it can be transformed into a rigourous theorem that goes like "If a property satisfies these conditions, then bloch's principle is respected".
I wouldn't qualify Bloch's principle as "most helpful", but it is certainly interesting.