Do you have your own source of income? Like do you babysit on a regular basis or have some other kind of job? Rabbits can get pretty expensive. Not only do you need to break down how much the monthly cost is, but you also need to figure out what all the “one-time” start-up costs are. You have to buy housing supplies and pay for your rabbit to be spayed/neutered. Plus you want to have an emergency fund or something in case your bunny gets sick and needs treatment. There are a lot of things that can go wrong that can be very costly to treat.

I say this because it sounds like your mom doesn’t want the added expense. If you cannot afford to pay for something, then the responsibility would ultimately fall on your parents to pay for it.

I agree with LBJ10. It is so great that you’ve done all of the research.
I got my bunny, Peter, when I was in 7th grade. I still have him, and I am now a junior in college. For about seven years, he seemed to be in perfect shape. However, this past year, I have had to spend over $700 on vet bills. The difficult thing with bunnies is that it can be really hard to diagnose what is wrong with them, so often, you have to spend a lot of money on just that alone.
I love my rabbit, and I will never regret getting him. He has helped me through a lot, but it is so hard to see him when he is sick. It is even more difficult when you almost don’t have enough money to pay for their vet bills. That has been the most difficult part for me. I want to do what’s best for him.
In seventh grade, I never would have expected him to get sick like he did.
So, as LBJ10 said, it is important to have an emergency fund.
As for convincing your mom, I think you are doing a great job. The best way to show her you can handle it, is by showing her how well you take care of other animals.

If you really want to convince her then –

Step 1) Keep taking care of the dog. If you and your dad promised to care for it when you got it and didn’t do that, then she has a pretty good reason to think the rabbit will be the same. Show your commitment by honoring your previous pet agreements, not just in the short term, but for the life of the dog. If she sees you aren’t just doing it to prove a point to her and then going to go back to how things were, she will see you’re serious about your responsibility.

Step 2) Save. Save. Save. Save up enough money to be able to not only afford the rabbit, but also have an emergency fund even before you get it. Rabbits CAN be very expensive. A single injury can run you $700-$1500 in vet bills. If you live in a place that requires vaccinations you will need to pay that every year. If you can not only afford the bunny and housing, but also have shown your willingness to have a savings account for potential emergencies, that will show you are financially able to care for the bunny.

Step 3) Be patient. Be respectful. It’s your parents house, your parents make the rules. It will probably take time to convince her, she probably wants to know it isn’t just a temporary project you’ll grow bored of. Prove your commitment to her and be positive about having to wait, don’t get upset with her over not saying yes.

Honestly, it sounds like you have a full plate already with your dog AND guinea pigs! Rather than stretching yourself thinner by adding an additional animal, I would recommend you spend your time and money improving the diet, habitat, exercise routines, and emergency vet fund for your CURRENT animals.

It’s awesome that you’ve researched rabbits, and I wouldn’t stop learning about them and filing it away in your mind in a “someday pet” folder! Your dog and piggies are (sadly) not going to live forever… and rabbits are not going away anytime soon! They’ll still be around in the next 5-10 years when you are self-sufficient, living on your own, and are ready for a new pet.