Hiya - Another young female here to toss you my opinion...
I got Guacamole Kangarilla 2 years ago. While I often panic about his health and post here about said panics, he's so far been a perfectly healthy, happy boy. I got him my junior year of college when I was 20. With this perspective, I will say the following:
Your mom may have any assortment of reasons not to want you getting a dragon. Unfortunately, for as long as you live with her, you kinda have to bin it and listen to your legal guardian. There probably is no way to "convince" her to let you have a dragon - but a mutually respectful conversation further investigating exactly why she doesn't want you to have one is a good idea. You might find out she has some misconceptions about the creature or a personal distaste for them, or otherwise that she has misgivings about your responsibility level etc... Finding this out her exact reasons is the first step in moving forward or resigning yourself to the fact that you're just not going to be able to have a reptile until you move out.
Personally, I would suspect one of the following objections:
-She's worried about cost
-She's concerned about mess/smell of Bearded Dragon or associated creatures (feeder insects)
-She's concerned about neglect or having to take care of it for other reasons
Cost you've already covered. You seem to think you can pay for a beardie yourself, so I won't touch on that other then to comment that you should be generous with your budget and make sure to have something set aside for "pet emergencies" - for instance my apartment had a power surge, so I needed to replace the lights for my beardie (roughly $80) as well as all my other animals (another couple hundred dollars).
Mess and smell is also up to you just doing basic maintenance and cleaning up after your pet, but insects are a big issue for a lot of people that you really need to consider if they are an issue for her. Crickets ARE going to escape, and if your mom hates them, this WILL be an issue. Roaches are easier to contain, but most people who don't keep insects hate them 100 times more than crickets. Most other feeders are either super expensive or don't provide full nutrition to your dragon.
Finally the last common concern - her ending up taking care of it. This doesn't necessarily speak ill of her opinion of you, btw... For instance, do you intend to go to college? A dorm is no place for a pet reptile. Regardless of the reason, if her concern is that SHE DOES NOT wan to take care of YOUR pet and that's her hang up, then my advice is to look up worst case scenarios. Finding a rescue that could take your pet in if you absolutely could not take care of it anymore might do a lot to ease her mind... But on that note, realize that you should never EVER get a pet with the intention of abandoning it.
To address somethings I have see on this thread...
Cats are dangerous. Everyone who says this is dead right... But that said many of them probably have cats and a good, responsible owner can avoid catastrophe. It is a big risk for a new reptile keeper, though.
Prices for a good set up are going to vary widely. The tank itself will run about $100-150 is you don't build it yourself, hit a really good sale, or find something suitable - THAT YOU CAN CLEAN - used. Lights and fixtures will run you about $100, with the cost of UVB being a re-occurring issue. Food will be fairly cheap, maybe $50 a month if you're not breeding your own insects and you're a little wasteful with greens. Electric should only be a couple bucks monthly. Vet cost varies widely depending on where you live. Hides and decorations also varies widely, but anywhere between $60-120 is what I would expect, again, buying new. That is a very rough and dirty estimate, but there you go. However note that now that I have everything set up well - I buy greens once a week and breed my own feeder insects - my reoccurring costs for Guac is about $5-7 a month, with about $20-40 every six months for UV light replacement.
I find some of the comments regarding your potential social life kinda... Condescending. Don't want to comment past that, but if you're feeling discouraged because of that, please know that I feel you. I'm new to keeping bearded dragons, but I've been keeping a multitude of other similar pets since I was 8, which is now a full 14 years of experience. I never found my passion for insects, amphibians, or reptiles to be a burden on me or me social life, and statements like that are incredible presumptuous of your personality. That said, it's presumptuous for a reason... People of your -arguably our - age group tend to be rash and not plan ahead very well.
Really think about what you're getting yourself into - a beardie is a 10+ year commitment... If you're 14 now and you plan to have children, for instance, there's a decent chance that your pet reptile will meet your kiddo.
Now, I am a full time student and I have a big bunch of pets that I spend about an hour each day taking care of and about $40 dollars a month, total. I didn't date seriously until I was 20, and I have been to exactly one of what I think most people think of when they think "parties"... I didn't like it. If you're the kind of person who really is good and ready for a dragon, then all you need to navigate is negotiating with your mother if you can, or using the time that you need to wait until you move out because your mom just absolutely won't let you have one building a bearded dragon savings fund.
You mentioned during sports season that you don't get home until late... One thing you could do is strategically get your dragon at the point of the year when you will have the longest amount of time before needing to stay late. Just a thought.
The best comment I saw here IMO was definitely Blackenedsabbath's. Take a measured, realistic look at why you want a dragon and if you're prepared to handed something that, if you take care of it correctly, will probably be with you when you are twice your current age. If you think you really want one after that, work on compromising with your mom... Either she'll let you get one, in which case yay! You get to have a beardie now! Or she won't let you get one... In which case, yay! You get to save up the money and resources to give your future beardie the best possible home. You might even look into getting a fancier morph at that point, for instance I am currently saving to get another bearded dragon of either a Red of wiblits morph.
Good luck!