Not a chance on a 121 revenue flight.

But if you do a 91 flight for repositioning or training it would be legal (and I've done it). It would be a very good idea to get your company's approval in writing first. The hard part here is knowing far enough in advance to get permission and then get your family member in position. Talk to your CP, and try to get advance written permission and then you could volunteer to do repo flights on your days off until the stars all line.

Make sure your airplane does not require flight attendants in order to carry pax. The generic 121 rule does not apply to a 91 flight, but if aircraft certification requires FA's that could be an issue.

Also you almost have to originate from a hanger or FBO, not a commercial terminal...there is no easy legal way to get your family through TSA for a flight of this nature unless. You could always list them for some other flight just to get a security doc, but that might not go over well if you got caught.