This is a question which has fascinated me for awhile. I fully agree with the OP, the selection of gorillas as humanity's third species to try to uplift really annoyed me, as did the fact that we hadn't tried anything easier to handle before moving on to chimps and dolphins.
When writing about uplift, I think that chimps and bonobos are interesting candidates since they're our closest relatives but have dramatically different social structures. So would geladas, which one could argue resemble grass-eating humans. Spotted hyenas also have a really interesting social structure which could make an uplifted version interesting if we didn't breed it out of them. If we were actually to engage in uplift, if makes sense to try to uplift species who would work or play with us in new ways. Uplifting apes would basically be making new species in our own image; I don't really see the point.
When it comes to self-awareness I'm a believer in the Machiavellian intelligence hypothesis: intelligence evolves when you need to remember your past interactions with other members of your species in order to decide whether to trust them. So the obvious candidates are species which already have a social structure which requires individuals to recognize and remember one another. Pretty much all species which display some of the cognitive skills we think of as unique to humans fit this pattern, and exceptions such as raccoons and octopi have relatives which are social.
This article has already mentioned that corvids contain New Caledonian crows, the only non-human animal known to be capable of using recursion to solve problems. Corvids also contain the only non-primate (and possibly non-ape) species known to be capable of reasoning about what other members of the group are thinking: if another scrub jay is watching, they will rehide nuts after the watcher has gone. Unlike with tool use in the New Caledonian crow, which naturally uses tools in its natural habitat, there's no reason to that theory-of-mind is unique to scrub jays. Parrots are the other particularly promising group of birds: the kea has been mentioned and is one of the fastest problem-solvers of any animal, and of course African gray parrots may be capable of forming semantically coherent novel sentences, which if Irene Pepperberg is correct is truly impressive: it's something even apes can't actually do. Many species of corvid and parrot are quite good at problem-solving. Members of the starling family are also pretty smart, and common starlings have the advantage of breeding really easily and eating anything.
With regards to mammals, vampire bats are the only animal species other than humans known to practice reciprocal altruism, prairie dogs have rudimentary language. Elephants are able to recognize themselves in the mirror and are exceptional in their memory (though it's not true that an elephant never forgets). Elephants would make terrible candidates to uplift due to their large size and long gestation period, but their relative the hyrax is also a social species. It too has a long gestation period, but if it could be reduced hyraces would make interesting candidates. So would the social mongooses, particularly banded mongooses, the only species I know of where all pairs get to reproduce and all members of the group help raise the young. Raccoons with their thumbs and their social relatives the kinkajous and coatis would also make interesting candidates, as would rats and prairie dogs. Sugar gliders, being tame, sociable, long-lived yet easy to breed, and surprisingly dextrous for a species without thumbs also seem like possible candidates.
If you're going to use primates, I think it makes sense consider thinking outside the apes. Spider monkeys problably are the smartest New World monkeys and have prehensile tails. So do capuchins, which are already are trained as helper monkeys for the disabled. Owl monkeys (another New World monkey), lemurs, and bushbabies are nocturnal, which we are not.
Now, if we're going to choose where to start, it probably makes sense to start which species which are social and either already cooperate with humans or have close relatives that do. That way you can claim that you have a good reason for breeding smarter animals. This makes dolphins certainly good candidates. The same is true of capuchins, which as I said we already have trained as helper monkeys. I'm skeptical of the claimed cognitive abilities of dogs, they seem to be savants at reading human emotions but not particularly exceptional otherwise, however our long association with them makes them good candidates. The mongooses humans have kept for snake control are not social mongooses, but I see no reason banded mongooses couldn't be bred to do all that solitary mongooses do, all that ferrets do, and more besides. Falconry dates back into prehistory and the most popular falconry bird these days (at least in the US) is the Harris's hawk, which also happens to be the only raptor to hunt in packs. Parrots can be trained to do tricks, are impressively dextrous, and might be bred/trained as helper animals much in the vein of monkeys. We might also potentially try to train octopi or cuttlefish to work with us in the ocean much as dolphins do, though I don't know how productive that would be.