I presume I'm odd, but I've never had any interest in toys, memorabilia or collectables, even as a child. DVD/Blurays, books and comics certainly, but I have no interest in owning a prop.
Having said that, if someone forced me to chose, I'd pick a phaser, tricorder or communicator.
Most of the years I worked backstage in musical theatre, it was on the properties crew. Over the years I had to beg, borrow, buy, and make an amazing variety of things, from junky stuff (West Side Story) to really expensive-looking stuff (The King and I), and deal with real food and live animals (The King and I; Gypsy). I remember getting into an argument with the director of Gypsy, trying to convince her that although the script called for a monkey, we couldn't use a monkey. She wanted to know why not. So I explained that the nearest monkey was probably in the Calgary Zoo and they would be unlikely to lend us one, even if we had someone who could care for it. Ditto for the goldfish - both of us on the props crew had cats, neither of us had cars, and we were not about to ask the janitorial staff at the theatre to babysit a goldfish.
So we substituted a dog and a rabbit for the monkey and goldfish. I discovered that although I love my own cats, I hated working with animals in the theatre.
I do have a few mementos of my theatre years, though - props that were so specific to one show that they wouldn't ever be used again, so rather than seeing them destroyed I saved a few. So that's why I have a caveman's club on my bookshelf, a pirate's dagger, some old-style telegrams, and so on.
I'm surprised no one has said Picard's fish. I was going to mention it because I'm a fish guy myself, but frankly that setup in the ready room has never done anything for me. I find Lionfish in general to be rather boring and I especially don't like bowl shaped fish tanks. They should have done a real corner tank setup in the ready room (a triangular or pentagon tank).
Speaking of fish tanks, doesn't Troi have one in her quarters as well? I seem to vaguely remember one, was it in the episode The Survivors that you see it?
I'd adopt the cat used in Tasha's hallucination (the episode where everyone was seeing things and people from their past, or fulfilling some fantasy). Of course I know these cats are dead now, but if I could go back in time...
Fish, though? Nope. Unless you're a breeder or some sort of tropical fish person, fish are really boring pets. You can't play with them, you can't cuddle them, they don't keep you warm at night, they need nitpicky care or they up and die...
