The pet rock craze was still going strong when I was a kid! I actually did own one myself for a little while. I gave it a name and a personality and everything. Turns out rocks don't make the most interesting pets though, because I got bored of carrying around a lifeless lump with me.
Shihtzufan, I remember Tamagotchi and those sea monkeys as well. I used to own both of them. I totally loved my Tamagotchi though and I looked after it for months before I ended up losing the poor thing. One time I had brought it to school with me and forgot it in my locker, and cried through the whole evening because I was so scared that it would die without me there to look after it. I never had much success with the sea monkeys though. I tried my best to get them thriving but despite the "absolutely guaranteed to grow" claim, no matter how closely I followed the instructions I wasn't able to get them going anywhere. The idea was neat though; I almost wonder if they still sell those kits, and if I would have more success with it now that I have better knowledge on taking care of pets.