I think creation/maintenance of a petsite would be quite brutal on me, as they are in essence quite ongoing and social communities, with some mind to user's experience.

BUT, I will imagine a situation where I'm just being paid to create the species, world, for a different team to actually manage the site and users, imagine all of the greater inputs... I am Simply the Art Development Team

HMHMHMM. Some basic things...
I would like it to SOMEHOW be not so monetary-based play as much as exploration-based. I always liked websites that involved a lot of clicking around, as a kid... I've never actually ENJOYED the act of 'making Neopoints', though I enjoy some amount of resource gathering like in fer.al. Consistency producing results!
On a lot of petsites, I often felt like the only way to gain a lot of resources was sheer willingness to participate with other users, but I'd really like a system that benefits the solitary user.

An ability to create 'scenes' rather than individual 'pet profiles' would be a key format change; various pets, bugs, plants, gathered under an umbrella defined by the user, encompassed by as many or as few individuals.
Whatever resources gathered could be allocated to a given 'scene', and individual items could be attributed to a given creature's 'things', and because I'm me, I'd like no restrictions on what can be a critter's 'things', even another individual animal.

I think with that basis you could create millions of narratives.... simple families, or collections, or darker things like a captor/captives, or even some sort of agricultural/farming situation. Any sort of enclave or cult. Interpretation is key!

Entirely feral world obviously, with no humans, no furries, no fairies. A handful of biomes, but I'd want to think critically about variable aesthetics and textures. I would probably like to imagine a lot of unique 'islands' or even collections of islands, archipelago, forming what you could think of as 'lands'.

Like for instance a plushie island, with entirely plush foliage, and different sub-areas of certain patterns and aesthetics (a more kiddish brightly colored area, an elegant royal patterned gold tasseled area, a scary ripped up patchwork area)...

There could be a goo/slime/jelly sortof area, with sub-areas that are... sickly toxic & dangerous... while another is silly Seussical whimsy!

NOW!!! another major element I would want!!! Is... you know how many petsites, have your 'pets', and then have say... 'petpets' or 'petpetpets' for Neopets,'Familiars' for FR, 'Rambles' for Livly... I want those sorts of creatures, BUT I WANT THEM TO ALSO BE CONSIDERED PETS!!!!!!!!!!!!

In real life, whether you have a pet snail, or a pet horse, those are simply pets!!
In the case of my own 'petsite', I would simply like no delineation to be made between a worm or a big animal, in terms of how the site handles them!!! WHAT I WOULDN'T GIVE TO HAVE SOME PETPETS AND PETPETPETS AS 'PETS'...!

Using the 'scene' profiles, one could display any creature in any way they so please, and ascribe any degree of characterization to that thing. I THINK THIS WOULD BE APPEALING TO LOTS OF PEOPLE!! Some people like bugs!! Tiny creatures are cute and fun!! IT WOULD BE DELIGHTFUL TO HAVE A PROFILE OF ALL LITTLE THINGS...! If you so pleased!

And I can't resist the Neopian urge to have alternate Forms of creatures, so any given critter would have an alternate form possible for any aesthetic.

Another thing is I want no straightforward earth-world things At All. No Apples and Apple Pie. Strictly fictional silly-named things!!! I feel most worlds only half-way do this... I want everything BESPOKE!!!

I've thought so long and hard about this all but I think that's all I can concretely say!!!!!! Hrmmm!!