Two other notes. First, not to gender stereotype, but that is a classic, masculine-style watch, and if it belongs to someone inactive for a long time, it most likely belongs to a male former Avenger?
Second, since Agents of SHIELD is not part of the MCU canon, what happened with Mockingbird there does not count, and in the comics Mockingbird and Hawkeye were married for a while. Maybe Marvel is doing another of their MCU mashups and Laura Barton is a retired Mockingbird and the Bobbi Morse version does not exist.