If your character could have any pet, what would he or she want? What would be most appropriate for the character, regardless of what he or she wants? In either case, why, and how well would they be cared for?
As a child Lume would have liked to have a cat, but couldn't due to her mother's allergies. As an adult she prefers not to get too attached to an animal, since if things go sour she might have to leave it behind to starve. She's considered getting fish, but felt they were far too boring for all the work involved.
As far as appropriateness, I (Eva) don't really think that most people have "appropriate" pets, since a pet does not represent an aspect of your personality but rather something you chose for companionship (or decoration if you aren't a very nice person). I _can_ tell you that it would definitely not be appropriate for Lume to keep any sort of awakened animal or plant, as she generally considers such beings to be morally above the line where ownership means slavery. If it can talk, it's probably not a pet in her eyes. She also isn't very comfortable with an electronic pet, since she finds such things annoyingly artificial. Come to think of it the closest she's got to a pet right now is Dezzy, and despite the fact that he's a rather dim bulb, she tries to treat him with something more akin to friendship.
Regardless of appropriateness or desire, if Lume had a pet she would do her best to take care of it. Despite her Mother's beliefs about her lackadaisical nature, Lume did learn a lot about responsibility growing up (she just spitefully ignored a good deal of it around her Mom). If another life depended on hers, she would try to make sure that she couldn't screw up that sacred bond with neglect, accident, or death. This would include contingency plans so that someone would takeover for her if she was incapacitated or killed.
Alternately she could just have one of those fern things that you can't kill short of lighting it on fire. Actually that's sounding remarkably good right now. All she'd need is a sign saying "This is a fern. If it looks dead, please water." and she'd be all good.