Do pandemonium and Pandora (she of the box) come from the same etymological source?
to answer your question, you can compare user7834's answer with this etymology
1570s, first mortal woman, made by Hephaestus and given as a bride to Epimetheus, from Gk. pandora "all-gifted," from pan "all" + doron "gift," from PIE base *do- "to give."
so the two words aren't probably related except for the fact that both words use the Greek word πᾶν (pan, "all")
Coined by John Milton in "Paradise Lost," Pandæmonium, from Ancient Greek πᾶν (pan, “all”) (equivalent to English pan-) + Late Latin daemonium (“evil spirit, demon”), from Ancient Greek δαίμων (daimōn, “demon”).