Definitely make sure the pet is not only good with children, but also "durable." By that I mean can put up with unusually loud sounds from a child and from the impulsive child wanting to touch. We had a dog prior to our autistic child... eventually our dog wanted to bite her. She couldn't tolerate her touching and loudness. After 10 years with her we gave her to a foster home. Later, when our daughter was older and understood better, we got another dog and things are working out fine. So, depending on your child, you may want to wait until they are older and can perhaps understand and demonstrate control around a pet.
I am a cat person and by no stretch a dog person, so if you love dogs you might want to stop reading because what I'm saying will likely tick you off :-) My son prefers cats, too. He's scared of nearly all dogs. One thing people really need to know before getting a dog is that you will have zero freedom and you cannot go out of town, or even stay away from your house for very long. If you get a dog train it to not whine or beg while meals are going on and absolutely don't feed it from the table. Dogs can be very destructive if they get bored and will chew things up. Nothing is safe in your house from a big dog, either. If you really think you need a pet and you've covered this with your family, trying fostering a dog or cat for a month and see what the reality is. Don't rely on memories from your childhood - chances are that someone was always taking care of the pet and you didn't notice.
We have 2 small dog here and my grandson loves them. I think having animals helps people in general. I recommend animals to anyone because they give unconditional love.
We have an 11-years-old orange tabby cat that we rescued when it was just 4 months. We neutered it as soon as we took it home, which made the cat very calmed. My son established a deep relation with it. They spend time together sitting next to each other and my son pets it. It has served to control his hyperactivity (ADHD) and seems to relax him. We discussed the idea since long before and were waiting for the chance to get a pet. I tested him prior with other cats in the rescue centers and pet stores. My son expresses deep emotional connection with the pets (cat and my sister's turtle). He cares and helps taking care of them, which allowed him to have responsibilities. I would recommend a pet to anyone, specially people with special needs and/or mental health conditions.
I wish people would not offer advice unless they are a paraprofessional or parent of a child on the spectrum