Seen on archives, everything form Bookbinders tape, gaff tape to you name it, to repair the hinges, but the posts are minimum three years old.
What is your favorite, and a source from which to get it?

Thanks for the help ...

Go with book binders tape as its probably more durable in the long run.
Gaffers tape will dry out in 7 to 8 years in a moderate humidity climate so in the desert it will probably not last as long.

I don't know that I would call it a "favorite", but I have used photo mount hinging tape, just because it was what I had on hand at the time. Seems to work OK, but durability may be a question.

OK got black bookbinder tape, and ready to do the job.

The original hinge tape is very secure, except where it is wearing, or has worn through at the hinge. Should I remove it, or attach the new tape to the old layer?

Remove it.
Use http://www.lowes.com/pd_16685-11743-407-45_0__?productId=3015285&Ntt=liftoff , motsenbockers liftoff adhesive remover, and remove the old tape and residue.
Goof Off is toxic, must be used in a well ventilated area;
Goo Gone leaves an oily residue that is hard to remove:
Krud kutter does not work as well but is better than nothing.