Michael J. Fox was born in 1961 as Michael Andrew Fox.

When he registered for the Screen Actor’s Guild, there was already a Michael Fox, so he had to insert his initial, which would, of course, have been Michael A. Fox.

…he hated the way “Michael A. Fox” sounded because it felt like he was referring to himself as one hell of a foxy guy. Incidentally, as a sidenote, actress Vivica A. Fox – whose middle name does start with A – realized the same thing but, unlike the man who would be McFly, loved it and decided it was a fun play on words that fit her sexiness.

The J in Michael J. Fox Stands for Andrew

It appears he may have taken the J as a tribute to actor Michael J. Pollard.