Since all three notions are closely interrelated, it's fairly trivial to take any one you choose as "fundamental" and define the others in terms of it:

- Sets as fundamental: define ordered pairs via Kuratowski's definition, define relations as sets of ordered pairs, define functions as relations not containing any two ordered pairs with the same first projection but different second projections

- Relations as fundamental: define sets as unary relations, define functions as relations the same way as before

- Functions as fundamental: define sets as functions where the codomain is some fixed singleton (characteristic functions are more appropriate for subsets of a set, rather than sets on their own), define relations as functions whose codomains are power sets

Because of this sort of thing, I don't think any notion of "fundamental" that's purely based on interdefinability really makes much sense.

I personally have a sense that some definitions are "constructive" and others are "destructive" (my use of "constructive" here is in no way related to "constructive" as in "constructive logic", it's just based on the English meaning of "construct"). Of the definitions I gave above, I perceive the definitions of functions and relations from sets, of functions from relations, and relations from functions as "constructive", while the definitions of sets from functions and from relations are "destructive". Since I can't think of any "constructive" definition of sets from functions or relations, while I can think of "constructive" definitions in the opposite direction, I would say sets are more fundamental than functions and relations. Whereas since both relations and functions can be "constructively" inter-defined, I'd say they're both on the same level.

Admittedly, I have no idea how to define "constructive".