I've been taking my pet out a lot recently. I even did a rescue with it, east of mid-pass well, when no fighter was available to help. So I have some suggestions on making them even more enjoyable.
a) Generate a "rank" sound when pets get stronger, like the ones when characters get stronger, except say reversing the tones so one knows which is which. Players get a kick out of hearing this, so I think it'd make pets more fun.
b) Generate a fallen message for pets just as for shared characters. I've had my pet fall out of sight and not know it until I left the snell.
c) Introduce a command to ask a pet to cry out. Dogs would bark; cats would meow; mice would squeak. This would be a /yell, and like that could not be repeated for some interval. Not only would this be cute, but it'd allow one to locate a pet one cannot see -- especially if it's fallen.
d) Have different pet collars available when they get much stronger, like fighter circle belts. Players like to show off, so this way they could show off their pets. (It could also be a coin sink.)
e) Allow creatures killed by pets count towards any challenge for the owner. We healers can't do many challenges anyways, because we cannot hit things. This would allow us eventually to try them all!
f) Allow pets to "stay" in a different snell than the owner. I use "stay" to heal up a pet, and it's very annoying when I forget to un-stay the pet and it goes all the way back to the stable. Also, there are many situations when it'd be better if I lured the next snell rather than have the pet enter with me -- the above rescue of mid-pass is an example.
g) Eliminate the 9-minute time limit on rescuing pets (fallen in a different snell). Instead, disconnecting would be sufficient to "revive" a pet in the stable, and would also send an unfallen pet back to the stable if one forgets or cannot get back to where one said to the pet "stay".
h) Allow one to push/pull a pet. This is probably the hardest to implement, given that one cannot currently click on a pet. However, it's a very useful tactic in the field, especially with a fallen, and so would help a healer practice it.
i) Allow healers of any ability to have one. I'm puzzled why this wasn't allowed when pets were introduced, unless the GMs were being cautious.
If the concern is rank acceleration, well newly exiled fighters have much more opportunity for rank acceleration than newly exiled healers. Such fighters can tag better than such healers, so their share is worth more. Groups are often willing to take fighters of any level as long as they can hit, but dislike weak healers because they both fall a lot (thus consuming healer resources) and reduce the number of shares for fighter cross-sharing -- especially if that doesn't leave a share for the senior brick. So if newly exiled healers can use pets to get stronger faster, I consider that a plus. It also keeps them interested in playing.
If the concern is that newly exiled healers aren't strong enough to support a pet, well that will simply mean more need for rescues! Especially if the time limit on rescuing pets is eliminated, groups could rescue pets that a newish healer cannot do themselves.
Having a pet to support can be a good way for a new healer to learn how to heal, as well as focusing their efforts on healing rather than on tagging.