My phone barely holds any battery. How can I convince my mom to let me buy myself a new one? ?

They are ripping you off. Wanting you to buy a new phone.

Search eBay for batteries. Do it yourself. Search YouTube. 

$30 at most

Get the battery replaced. Go to a more reputable shop that won't charge so much. Seriously your parents are not bank accounts, so quit being so unrealistic. Learn to take responsibility, because it's going to affect the rest of your life if you don't.

Never take anything for granted, because YOU WILL lose it all.

Battery replacement $400? 

Nonsense! It doesn’t even cost that much on a brand new iPhone 12 PRO Max. Even an iPhone 8+ battery changed by an approved Apple repair shop only costs about £60 (about $83 USD). An unofficial repairer using an inferior battery for that model only costs about £35 ($48.50). For just £15 including shipping I can buy a battery complete with fitting tools and do the job myself.

It’s even cheaper on the Galaxy S9: I can buy a self-fit replacement battery with shipping and fitting tools for just £6 ($8.30). It’s also technically an easier job on the S9. I’ve just looked online at a local shop to me and they replace the S9 battery for just £25, including a warranty. That’s just $35 USD.

US repair prices are also usually cheaper than in the UK.

So if you want to persuade your parents to let you just pay to get the battery changed, the answer is easy. Stop using the duff battery as an excuse to try and get a new phone and stop treating your parents like fools.

I DON’T suggest that you try the do it yourself replacement unless you have a little bit of experience: YouTube instructional videos always make the repair look easier than it truly is. It’s not exactly difficulty but it it very easy to ruin the phone if you lack experience and make a mistake.

My guess is your mom doesn't have the credit or want to take out a loan for $1000+ so you can have the latest phone to impress your friends. She's on the hook when you can't/don't pay because you needed money for other things. 

The S9 is still good phone and sells for $200 used and battery replacement is only $70, so you are very wrong about battery replacement bring more than the value of the phone.

You are also wrong about prepaid not having enough data.  Is unlimited data too little.

You have the option of buying a refurbished phone that is a year or two old for much less than new.  Look on Amazon.

There are gray market new phones from reputable Chinese manufacturers on websites like Gear Best and Ali Express.

I know you are only looking for answers you want to hear and not the ones that will serve your situation best as can be seen from your rash if thumbs down.

Replace the battery it saves you lots of money!!

you do not need a new phone just replace the battery

Find out if your mother insured your phone and if the plan covers bad battery then for whatever the deductible cost you should be sent a reconditioned phone your model or new to last you until 5g is everywhere and you have a reason to buy an new phone likely in a year or two. 

The S9 is a decent phone.  Just take the phone to a local cell phone repair shop and have them replace the battery like everyone else does when their battery goes bad.  

Use the recent kidnapping and murder of a woman walking home in London as a reason why you need to be able to contact her when you're out, if you want to go 'big'.

If you are planning on getting a new phone I would get an apple. I personally think they are way easier to use. I have an iPhone 11 and I love it. My battery lasts for a long time. I also got a deal on it so I didn't have to pay the full price. I would write her a letter or a text explaining on why it is important for you to have a new phone. I'm sure she will understand, especially that you are paying for all of it. You could also ask her what you can do to get it.