Surely there are minimal subsets of functionally complete operators, in general different ones for different three-valued logics. E.g., for Łukasiewicz's logic (L3), ∧ (conjunction) and ⟶ (implication) will do (implication is not definable using other operators).

I doubt that you could define, L3's operators just with NAND (or NOR). In L3, "N implies N" is T, and "not N" is N (where N is the third truth value). So, in one case, a formula whose components are all evaluated with N has truth value T, and in the other has truth value N. This is not a proof, of course, but I'd like to see such a definition of NAND :)

Interestingly enough, with F571 SQL's propositional fragment is functionally complete: any three-valued truth table can be defined. For instance, Łukasiewicz's P ⟶ Q can be expressed with

    (not P or Q) or (P is unknown and Q is unknown)

and the Słupecki operator, which is needed to obtain a functional complete system from L3, can be expressed with:

    (P or unknown) and unknown