Been there.
Exact same thing happened to me. Exact. Work let me upgrade my MacBook to a MacBook Pro, I was swapping the 8gb of ram from the old one to the new one, apparently the chips didn't go in all the way into their sockets and got stuck at the edge near the battery. Used a screwdriver (Phillips first) to try and pry it out, that broke off the same component you broke off. Then tried a flathead to get the corners while pushing down and letting up in between. Well once it finally popped out, the ram was bowed, which was easily bent back in to place. It all shows as working and at full capacity, but I don't trust it. I ordered a new 8gb kit from amazon and had it rush ordered. Should be here tomorrow actually.
My assumption is that the component that broke off wasn't extremely crucial. Obviously important but not to where it won't function without it.
Perhaps neither of us should use screwdrivers for that purpose again...also, be sure the ram is in all the way. You shouldn't see much of the pins showing (if any at all) if they are fully in. Basically, just double check your work. That goes for everyone.