LOVE Laura U's answer.
But I agree with the other posters here.
A 3 year old. He is a boy. Those 2 combined, make for a pet that is going to be handled... not gently.
Unless he is very good with impulse control... and being gentle.
Still, at this age, they will not automatically know how to handle a pet.
YOU will be the one to care for it and referee and supervise.
I would wait, until he is older.
When my son was 2 and my daughter was 6, we got a rabbit. But it was for my daughter. My son, in his 2 year old mind, understood and 'we' called it his too... but for my daughter. But, he, was pretty good about handling the rabbit. But, boys are really rascal. My son, would put toys on top of it, or hats etc. and tug its ear. To him, innocent stuff. To us/my daughter/the rabbit... EGAD! He did this, when I was right there. Little kids and their hands, move fast.
Luckily, the rabbit we have, does not bite. Some do.
And they don't make noise.
Guppies, are real easy. But you cannot play with it.
But they are easy.
Maybe a Guinea Pig. We had those too. But they do make noise... whenever the refrigerator opens. But they are cute, have pellet poop you just sweep up and live in a cage. They eat lettuce/veggies. Stuff you would normally buy for your own consumption. Nothing exotic.
I would say, a fish.
Or a Guinea Pig.
IF your son, is very good about his impulses AND he is gentle, AND he is not a rough-houser... then maybe.