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

No. Baron Nashor's aura gives other friendly units +2 Might, so your 3-Might unit enters your Base already at 5 Might — it never exists on the board with Might below 5, so there's no moment where it "becomes Mighty" (Rule 709). This is the same principle as playing a unit into Trifarian War Camp: the passive applies as the unit enters, with no window for a "becoming Mighty" trigger (FAQ #5835).

If the unit were already on the board and then gained the bonus (e.g., Baron enters later, or you move an existing unit while he's out), it would become Mighty at that moment. And if Baron leaves the board, the unit drops back to 3 and stops being Mighty.