formal framework for talking about 'minimal couplings' usually on physical theories one would have Lagrangians or Hamiltonians with multiple fields; say, a vector $A_{\mu}$ and a scalar $\phi$ and one would postulate ad hoc a coupling between the fields which will be used to derive some physical states and then, retroactively support the coupling from a match with the physics
Usually one reads about 'minimal couplings' and for scalars with vectors on might see a factor like  $\frac{ \partial{\phi}}{ \partial{ x_{\mu}} } A_{\mu}$ 
But my question goes on in what general or abstract sense such factors represent a minimal coupling? minimum of what? if is the total degree of powers of the fields? do we have some variational principle for lagrangian couplings that these terms can be called a stable point in such variations?
 A: They're the most renormalizable (or least non-renormalizable) couplings. If the term in the Lagrangian has a coefficient of dimension ${\rm length}^\Delta$, we want to minimize $\Delta$.
Equivalently, we want to minimize the mass dimension of the operator. That means reducing a combination of the powers of the fields, as well as the number of derivatives.
For example, $(\partial_\mu \phi) A^\mu$ is the minimal coupling between $\phi$ and $A_\mu$ (among the Lorentz-invariant terms) - this bilinear term is usually banned by gauge invariance or conservation laws (symmetries).
The dimension of the operator is 3 - one from $\phi$, one mass from $\partial_\mu$, one mass from $A_\mu$. Every factor of $A_\mu$, every new derivative, or every new copy of $\phi$ would increase $\Delta $ by one away from the minimality.
The points above are kind of modern - they are related to the Renormalization Group of the 1970s. The minimal couplings are those that are most important at long distances (by power laws) and least problematic (divergent) at short distances. During Einstein's times, the justification of the minimality would be understood just heuristically, based on feelings of beauty etc. That's no longer needed.
Of course, one could consider non-polynomial or even non-local interactions for which the discussion above would break down - but such terms would almost certainly not be called "minimal", anyway.
