Cheerdog, i have to say i do agree with your mom. I have been there before. You are in so much pain right now, You need to let yourself grieve. If you get a dog (puppy) too soon you may end up regretting it and trust me you don't want to try and "replace" your old dog with a new one. The time WILL come when it is right to get a new puppy, let Everyone in your family have a chance to grieve.
I agree with with klkrieg. It was over a year after my last dog died before I was even tempted to think of getting another one. You should let yourself and your other family members grieve over the loss of your old dog. If your dog was in the family for a long time, it's just like losing a human family member to some people and it may take some time before they are ready to let another pet into their hearts. Don't get a new dog, just to fill an empty space.
I have been thru this several times, sometimes it took me a year or so before wanting to get another dog and there have been times it was only 3 mos. before I was ready to get another dog. I don't know why it took so long at times, I loved all my dogs equally, but I knew when the time was right, it felt right. Allow yourself and your family time to grieve and trust me, you all will know when the time is right. Don't rush it, specially if you are just trying to prove something to your mom, that is diffently the wrong reason to rush into getting a new furbaby for the family. Time will help the healing.
Give it just a little time. When the time is right I'm sure you'll get a new puppy. Look forward to it and give your mom sometime to get over that your dog just died. She wont forget about your needs. I'm sure she is a great mom.