Who Wants It?
When giving away your great-grandmother's pearl earrings or weathered recipe book, or a painting that came to America with grandparent immigrants, or the shadow box of tags belonging to Sparky I, II and III, you'll want to start with one basic consideration: Does someone want it?
If you have a relative who has expressed noteworthy interest in a particular heirloom, that's probably where it should go. Another consideration is emotional attachment, either to the heirloom (perhaps it lived in someone's childhood bedroom) or to the descendant who passed it down (who was closest to Grandpa?). If you give the heirloom to someone else because, say, you think it would work better with his décor, the result could be hurt feelings. Possibly longstanding ones.
If no one stands out as the obvious designee, or, much hairier, more than one person stands out, you have some potentially difficult decisions to make.