At the prison in The Walking Dead (2010), Hershel (Scott Wilson) gave Glenn (Steven Yeun) a pocket watch that had been a family heirloom. Glenn initially took it as little more than a nice gift, but he soon started to see the larger significance of the gesture. Hershel had truly accepted Glenn as the man who could protect and love his daughter, and bestowing a Greene family heirloom was his way of accepting him into the family.

An analysis on Team Durango puts it well:

“It’s more than just a watch because of all that possess; it’s a time-keeper, a record of ancestry, a history book. Hershel’s watch holds an account of not only the present but also the past, before the infestation, which is why giving it to Glenn is such a big deal. Even under normal circumstances a gesture like that would be powerful, but in this situation it shows that Hershel is willing to put his daughter’s life in Glenn’s hands in a time of extreme danger, which amplifies its significance tenfold. In that exchange, Glenn proclaims his loyalty to Hershel and his family as well as accepts the burden of responsibility.”

It’s also a reference to the way times have changed - and the fact that each character knows their time to survive is running out. None of them expect to live to be 80 and comfortably retire. They know as the minority in the world, they are on borrowed time with each day they exist, and the pocket watch serves as a reminder of how precious their moments together are.