If I play a 3 might unit to my base while controlling Baron Nashor does it "become mighty" at any point?

No — the unit is never sub-5 Might while on the board, so it never "becomes Mighty."

Baron Nashor's passive gives other friendly units +2 Might and applies anywhere on the Board, including your Base (Rule 365.1; the Board includes Bases). So the 3-Might unit enters your Base at 5 Might immediately — there is no window where it exists in play as a 3-Might unit. Per Rule 709, "becomes Mighty" requires a Might change from below 5 to 5+ while the unit is in play, and that transition never occurs here. This matches the Trifarian War Camp precedent: a unit played directly into an aura that pushes it to 5+ does not trigger "becomes Mighty" effects, though moving an already-in-play unit into such an aura would (Rule 709).