You're trapped. Responsibility includes dealing with failure - which means, if you give him full responsibility, the pet will likely die, which will be a conflict with you. A cat may actually be the least problematic choice, as cats are pretty good at caring for themselves - but in any case you (I assume you are the parent/guardian) will have to deal with the fallout if the cat is injured or simply bored.
Susan (a fantasy figure in Terry Pratchett books) deals with this problem with respect to the classroom pet by choosing pets that a) are easily replaced when they're not looking (hence the three meepos - gerbils that were replaced without the kids being aware) or b) hard to tell whther they're alive or not (stick insects as the last resort).
For your five year old as sole responsible - won't work with any pet that might have to be brought to a vet (so no mammals) or requires an extensive technological life support (so no fish). You're tuck with invertebrates. An ant farm migth work, or a small freshwater snail tank , technology free (ok, a light for the plants may be feasible).
PLeae, no Goldfish. Yes, they can survive in a (too) small tank for a long time. They're simply hard to kill. But not for a five year old who wants to take them out of the tank...