In most cases, closely-related family members share a last name. Choosing a tasteful presentation of that name is a good way to tie a family together by way of a tattoo. The name can also be combined with accompanying artwork or a character that represents the family name or the ties that bind it. Some family's names are derived or shortened from ancestral origin, so using the ancestral name could also be a nice presentation of the family moniker. A family crest is also a wonderful way to show pride in the family name.
The tragedy of losing a loved one can bring a family closer together. Remembrance of that relative is a wonderful way to honor their impact on the family member's lives. Their name, dates lived, and a meaningful character or artwork to represent them would make for a good, concise tattoo. The artwork could be inspired by an interest that person had, or something that symbolizes their spirit.
Some family members pursue common interests, lines of work, activities, or hobbies. In a time when having your favorite sports team tattooed on your body is considered normal, it would be more than socially acceptable for a family of iron workers to have a shared hammer and anvil tattoo, or a family of sky diving enthusiasts to have a parachute tattoo. Finding common ground for the tattoo is the most important deciding factor.