In the 1970s it was common (in my area) to hold baptisms that were not part of an assembly or convention; they were held at someone's house who had a swimming pool. My brother, no longer active, was baptised at one of these events.
That practice was stopped for a reason that I'm not clear on.
This leads me to think that a family member probably would not be allowed to baptise a new one, but that's just a gut feeling, and I realize it might vary from place to place in the world since organized assemblies may not be as common everywhere as they are in the U.S.
Added note: These baptisms-on-the-side were done in the early 70s, and may have been a result of the 1975 end-of-the-world expectations being promoted by Witnesses. After all, the end was so close that 'there might not be another assembly,' so better get them baptised as soon as possible. That way 'they can be in the New System with us, dontcha know.'