I often talk to my girlfriend about the 'big 12' mammal species. In short, the three genera of great ape (orang-utans, gorillas and chimps), the other species listed in the opening post and seals.
My theory is that a zoo needs half or more of these species to pull in and sustain the crowds. Werribee Open Range Zoo, for instance, has only five of the big twelve mammal species. Werribee is actually struggling to maintain the boost in attendance that was achieved following the development of the Pula Reserve. The 2010-11 acquisition of gorillas, however, is expected to create a significant boost in appeal for the zoo. It's not a 'new' animal as such, since Victorian zoo-goers have been able to see gorillas at Melbourne Zoo for decades. But the addition of a great ape species will cause many casual zoo visitors to consider Werribee a better-value day out.
By the time you're getting to having three-quarters of the big twelve mammal species (as Melbourne Zoo does) you're starting to have what many zoo visitors would consider a world-class zoo experience. Melbourne has, only a month ago, gone from having 8/12 on display to once again having nine, with the recent re-opening of a seal exhibit (like others above, I don't really count pygmy hippos as having the same impact, although they fall into the "important smaller species" category mentioned below). I expect that 2010 will see a considerable boost in attendance over 2009 as a result... time will tell whether I am right.
There are other animals, as the thread starter correctly points out, that have an 'x-factor' appeal. Giant pandas are expected, I think I read, to double Adelaide Zoo's visitation in 2010. Dolphins are also in this category.
There's one warning I want to sound though. It isn't enough to have a collection made up almost exclusively of the superstars of the zoo world. I also think that having a good support cast of (mostly) smaller, often more active species - such as otters, meerkats, gibbons, monkeys, parrots and penguins - is critically important. I also think that a reptile house, walk-through bird aviary and even a small aquarium (ideally with sharks and rays) are important factors in creating the 'complete' zoo experience for casual visitors. Here in Australia, a zoo without the more familiar natives (kangaroos and koalas, and to a lesser extent wombats, platypi, echidnas and Tassie devils) will struggle to get international visitors interested.
Rarities that are obviously distinct from more commonly seen species will have an impact. Species or sub-species that might be unique to a zoo, but which aren't visually distinct, I doubt have any impact at all. We on Zoochat might get a thrill out of seeing 10 different antelope species, but they'll be ignored by most visitors.
I visited the National Zoo and Aquarium in Canberra yesterday. It's a small zoo with only about 100 mammals and maybe 200 animals, other than fish, all up. But it has a collection that ticks just enough boxes - lions, tigers, bears, giraffes and zebras, with the 'x-factor' taxon of sharks - along with a few charismatic smaller species (koalas, otters, monkeys, cheetahs and snow leopards) to get by. Having said that, if I was the owner I'd be itching to get some rhinos or chimpanzees, as the most obtainable 'superstar' species that could put me over the magical 6/12 figure.