I mean yeah a min raise is yet another variable (just the raise size in general).
If we think 3x is a close fold, maybe a min-raise is a close call. Depends though, if you feel comfortable postflop. For example, vs. some regs, if the flop comes Q72r, we check, he c-bets and we call, turn is 3, if we check and he bets again, vs most regs this can often be fold in these positions, as his range is often going to be AQ+, and our hand (AQ) is crushed by that range. He bluffs 0, and value-bets nothing worse. So if you can think about these kind of things, lean towards a call.
Basically lets say we are pretty sure when a specific villain 3xs UTG, we have a BREAK EVEN call in the SB with AQo. Well.. consider what variables there are.. Villains range, effective stacks, size of his raise (therefore what pot odds we are getting), our ability post flop etc. If any variable changes, it either makes it more of a call, or more of a fold.
For example, if instead of the specific villain, we are now faced against a really loose villain, we should now lean towards a call, as AQ does better vs a very loose range for obvious reasons. Similarly if instead of 3x, the same villain makes it 5x, we should now lean towards a fold, even if his range is the same to open, as we are getting considerably less favourable pot odds than previous etc etc..