Is the sentence
a correct translation of
One of my family members is thin and cute
In this case, "私の家族の一人は" is correct. Although it might feel more natural to specify which one(brother, sister, father, etc...).
However, the following change should be made :
痩せて > 痩せていて
Adding the "いて" is needed to reflect the ongoing state of being thin rather than the plan or the action of loosing weight.