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I come from a background of having a few lawyers in the family. Is using this information an effective way to somehow show that I have what it takes to make it through law school and become a lawyer, or am I wasting the admission officers time?

Most likely wasting time unless it's somehow part of a compelling and interesting story, which I doubt.

Waste of time and will hurt you unless they're alums of said school, in which case you could mention them as part of the reason you want to attend that school (but not for the reason you stated, which suggests that you should be judged on your family's merits rather than your own).

Agreed. Waste of time and a turn off ... schools really don't even care if you are a legacy unless you meet their profile.

Turn off.
Schools assume that some people are lawyers because their family members are.