Where did the name Phage come from ?

Well a "phage" in biology is something that eats/devours... but I'm not sure that actually has anything to do with the item Phage.

"Phage" is a virus or bacteria, iirc. I assume they called the item Phage because its original on-hit effect was the reduction of enemy movespeed, which is sort of like weakening the enemy with illness. Now, of course, Phage doesn't reduce enemy movespeed anymore, Riot changed it so that Phage increases the owner's movespeed.