My parents have long been rabbit folk. Our family has had a total of three rabbits, all when I was a kid. I think the last one passed away around 2010, after making it to the impressive age of 13 or thereabouts. I was never particularly close to the rabbits, but looking back, I'm pretty sure I appreciated their presence in my life. They were like one of those things you take for granted until you've lost it, such as the ability to walk or to breathe normally. They demanded a lot of time and attention, but so does every worthwhile endeavor in life.
Pets are of philosophical interest for multiple reasons.
1. They can give us a sense of purpose. There are several ways in which they might do so. First, since they are incapable of caring for themselves in a domestic environment, they––like children––depend upon us for their survival. This gives us a reason to get up and get going in the morning, rather than lying in bed and wondering what the point of it all is. Second, once we have a good relationship with them, they often make us feel we are accomplishing something of real value with our lives. Finally, they can bring us great joy and happiness, which will incline us away from the view that life is meaningless.
2. A genuine relationship with pets seems quite feasible, despite the difference in cognitive abilities between pet and owner. I don't know all of the necessary and/or sufficient conditions for having a relationship with someone or something, but I do know that a relationship must involve reciprocity: it can't be unilateral. So as much as someone with a beloved pet rock might protest, one cannot have a genuine relationship with a rock because rocks can only react to what we do with them; they cannot instigate actions, much less direct actions towards their owner. By contrast, one can have a genuine relationship with, say, a lizard, although it might be difficult to decide at what point the lizard acts with its owner in mind, rather than acting only as though it had its owner in mind.
3. They can, in theory, teach us virtues such as patience, commitment, and charity. Patience might come from learning to tolerate the sometimes destructive habits and peculiarities of the pet; commitment might come from having to tend to its daily needs; and charity might come from appreciating the unconditional love shown by it. Of course, being a responsible pet owner won't always, or even often make one a better person. Hitler was quite fond of his German Shepherd Blondi, for example. But the potential for self-improvement is arguably there.
4. They connect us to nature. As C.S. Lewis explains in Chapter III of The Four Loves:
[The] higher and domesticated animal is, so to speak, a "bridge" between us and the rest of nature. We all at times feel somewhat painfully our human isolation from the sub-human world--the atrophy of instinct which our intelligence entails, our excessive self-consciousness, the innumerable complexities of our situation, our inability to live in the present. If only we could shuffle it all off! We must not--and incidentally we can't--become beasts. But we can be with a beast. It is personal enough to give the word with a real meaning; yet it remains very largely an unconscious little bundle of biological impulses. It has three legs in nature's world and one in ours. It is a link, an ambassador. Who would not wish, as Bosanquet put it, "to have a representative at the court of Pan"? Man with dog closes a gap in the universe.
I see no reason to restrict these remarks to just dogs and other "higher" animals; any animal should do. The important point is that pets represent an intertwining of human history with the history of the natural world as a whole.
5. In a similar vein, pets remind us that the world doesn't belong to us: we share it with countless other living things. And, as the most intellectually advanced of all animals, and thus the ones with the greatest power to change the environment, we arguably have a responsibility to care for the world and the many creatures that inhabit it. This is one way of understanding God's injunction to mankind in Genesis 1:28 to "have dominion over the fish of the sea, and over the fowl of the air, and over every living thing that moveth upon the earth." Now, this doesn't mean we ought to preserve and cherish everything that's alive, full stop. Mycobacterium tuberculosis is alive, as are mosquitoes, venomous snakes, and poison ivy. It just means we ought to be good stewards of the natural world, and to be mindful of our relationships with our fellow creatures. This seems fully compatible with wiping the smallpox virus off the face of the earth (or at least confining it to a handful of infectious disease laboratories), or seeking to curtail the population of mosquitoes, guinea worms, or other living organisms which pose a real threat to us or other species.
6. They show us the stark difference between a creature that is tamed, and one that is wild. The typical dog, for instance, is either docile or prone to attacking only those who are hostile towards its owner, whereas the typical wolf won't hesitate to tear anyone apart if it's hungry. One of the principle tasks of human development is, in a way, to become more like dogs, and less like wolves. We can never fully escape our animal instincts, but we can prune and redirect them through education, habit, mimicry, and other means.
7. They serve as a reminder of our inevitable decline and death. Most pets don't last more than a couple of decades. Some (eg, hamsters) only make it a few years. When their entire lives play out before our eyes, they––like a sort of living memento mori––serve as a reminder of what will become of us in due time.
Note that this is not meant to be a list of reasons for owning a pet. Whether to get a pet is a highly individual decision which will depend more on practical matters than philosophical ones. These are just some things we observe when examining pets through a philosophical lens.