Why is Stitch (in the side), called Stitch?

Answer 1

An old meaning of "stitch" was a prick or a stab of a knife or a sharp instrument. It went on to mean the sharp pain caused by such a stab and from that to a stabbing pain in the abdomen or thorax which is its modern meaning.

Answer 2

Stitch is one complete movement of a threaded needle through a fabric or material such as to leave behind it a single loop or portion of thread, as in sewing, embroidery, or the surgical closing of wounds.

Thus anything that involves a needle and a thread (fiber or synthetic or even a nerve tendon), is called a stitch