A garment or other item that has been passed on from a father to a child. A hand-me-down, but specifically from dear old Dad.
Misha says, "Hey, cool Hawaiian shirt!" and Travis replies, "Thanks, it's just a dad-me-down."
To come to the realization that you are ready to settle down. To grow up and accept responsibility. To man up so to speak.
Bro, I've been doing a lot of thinking lately, I'm ready to Dad down...