Remember the generalization rule: Factors whose effects you would report and interpret are fixed and factors you generalize about are random. So experimental units are mostly random factors. If it is the only random factor, a fixed effect model suffices (in fact, the randomness of this only factor is the reason why you need statistics).
In your case, you seem to have two random factors: The individual effect of each person and the "family effect". You don't want to report the names of the families and the individuals as informative, so the generalizsation rule says it's random.
If you would omit the family effect (allowing a fixed model), you would claim that the health outcomes of family members are independent. Nobody would believe this.
So yes, you need a mixed model. The next question is however, if you want to model the family effect as uniform in your sample or as depending on some covariate.