First of all, understand that this will be your dog. Your kids will walk it for the first week, and only if it's not raining, snowing, hailing, steaming hot, or early in the morning. Do not believe them. I know they are wonderful kids and they are totally sincere right now but really, there are no parents on the face of the earth who will tell you this will be the kids' attitude for very long. While they're whining about how they're tired or they'll do it when the TV show is over or that it's the other sibling's turn, the dog has now peed or pooped on the carpet. Oh, and the novelty of picking up dog poop wears off after the first time!
Consider going to a shelter, particularly one that does a good job of getting the background on the animal. Most dogs in shelters have not been abused. I wouldn't get one on line or from a rescue organization (even a good one) if the dog has to be transported a long distance or flown to you. There's less support for you, and you really don't know what you're going to get. A local shelter knows they'll see you again, and the dogs haven't been moved around so much and are less traumatized. They should be asking you a lot of questions (we did a 5-page application!) and working hard to make a good match. Do not buy a dog at a pet store. Ever.
Consider a dog other than a puppy - a 9 or 12 month old dog is probably housebroken already, and is old enough to go all night without having to go out. A puppy, like a human baby, can't go all night.
Consider a mixed breed - thoroughbreds are expensive. There's no such thing as a truly hypoallergenic dog, but Tibetan terriers, Maltese, poodles, Portuguese water dogs, and some "-oodle" crosses (e.g. a schnoodle, cross between schnauzer and poodle) often get the no-shed label. We had a Cairn (purebred) which shed hair but didn't have dander, so people weren't allergic to her. We now have a rescue dog that is part Cairn, part Affenpinscher and part who-knows-what but it seems to be terrier breeds, who does not shed AT ALL. Nothing. I don't really know why, but work with the shelter to express your needs, and be prepared to wait for the right dog to come along. In general, smaller dogs live longer, by the way.
Barking all night - stay away from the hounds (beagles, foxhounds, etc.) because they love to bay at the moon or bark if they hear an animal outside at night, but remember that any puppy can bark for comfort, like babies cry. Also be sure to spay or neuter the dog - unneutered males always want to go out at night in pursuit of females. They also don't always have good dispositions.
You want a small dog, but don't get something really fragile because you have a toddler. Even a big dog like a greyhound is not good with toddlers - they are docile, but have fragile bones and a tumbling toddler can break a dog's leg. Terriers are great family dogs but have to be leashed - they are bred to take off after rodents, so you can assume they will chase every squirrel and chipmunk. But we've had terrific luck with them.
Do not get a Dalmation - they aren't good with kids. No matter what you do. They don't even like adults all that much LOL!
Plan on crate-training your dog for a lot of reasons. They need a place to be secure when you go out or travel, they need a place to get away from a 2 year old, and they tend to sleep better and housebreak better if you can confine them someplace where they will not pee. There are plenty of DVDs on dog training - borrow some from the library to get an idea. Our shelter told us absolutely NOT to follow the "Dog Whisperer" - they do not endorse his methods at all. So get some info from vets and dog experts.
Finally- BUDGET. For food, vet care (emergencies, illness plus routine and neuter/spay), toys, monthly heartworm meds, flea/tick stuff, grooming, annual license, 3-year rabies & other shots, fence/lead/run for outdoors, equipment (bowl, leash, crate), boarding during vacations,and training unless you absolutely know what you are doing. It adds up.
Hopefully you will find one your mother can deal with. If not, leave the dog at home. Do not adopt a dog and then give it back because your mother cannot be around it. It's not fair to the animal. Every dog should be purchased or adopted thinking it's going to it's "forever home." It's not like returning a dress that doesn't fit. If your mother is that important in the scenario, then take her with you. If she can't go into the shelter, then bring the dog on a walk away from the others and greet her.
Any little dog needs to be an indoor dog - there are coyotes all around. And outdoor dogs are often neglected and a nuisance to neighbors if they bark all the time. But you still are going to walk that dog and exercise it and play with it.
If you do it right, dog ownership is wonderful. Our Cairn died at almost 14, and gave us so many wonderful years and laughs. We waited 2 years, and my husband still didn't want another dog because he was still missing her so much, but then I saw a blurb in the local paper listing all the dogs the local humane society had for adoption. When I saw this young part-Cairn dog up for adoption, it spoke to me. What can I say. My husband dragged his feet, but after a week, he was completely smitten with this little adorable critter who had been abandoned by 2 previous families. Took us some patience and some good understanding of her temperament but she was great with us within 2 days, and good with outsiders within a month. Not bad for all she had been through, and now she's a real joy.
And did I mention, it's YOUR dog and you need to walk it because your kids won't??!! LOL.