I'd suggest the easiest fix would be to replace the screen door lock completely. I recently had a similar issue and found it was an internal part that was broken. There are no internal replacement parts readily available. With a reasonable level of D.I.Y. skill, you should be able to replace the lock in 30minutes. There's nothing particularly complicated about it. I didn't bother reading the instructions, as it is all fairly self-explanatory. Just remove all the screws on the current lock and follow the same procedure in reverse to install the new one.
I just took the broken lock with me and found the one which resembled it most. The Rolltrak Black Hinged Screen Door Lock looks very close and can be used to retrofit existing similar locks.