Would it change anything in your mom's mind if you kept your rats and their home in your room, completely out of her sight, and she never even had to encounter them?
My mother is afraid of rats, but as long as the cage is in my room, and I care for them, she's willing to let me have them.
After getting to "know" them a little, she even says that they are "cute" - as long as they are in the cage.
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I don't know if this will work in your case, but in my case, I just kept bringing the idea up every once in awhile, and eventually, my mom came around to letting me have one, and then two. I wouldn't be pushy, or anything, but just every once in awhile, I'd mention it. I'd include the pros and the cons, as well, so that she wouldn't think I was only seeing one side of it. Over TIME, she was able to see that I was truly serious about wanting a rat, and that it wasn't going to go away.
For example, I might say, "I read about a pet rat that was running loose around the room, and chewed through a cable. He's lucky he didn't get electrocuted, but still, now the family has to replace that cable. Of course, rats shouldn't be running around loose in rooms with wires and other dangerous things. I know that I'd never let my rat get away with something like that."
Then drop the subject for a week or two. But ~do~ refer to "your rat" - best to start by referring to one, even if you plan to get two - so that, over time, your mom will see that you are truly serious about having a rat, and that you do plan to have a rat someday, even if someday is when you move out of her house.
She knows you're on a rat forum, right? This can be a good place where you "read" about "no-no's" that you can discuss with her:
"I'd never let ~my~ rat's cage get messy. It's not healthy for them."
"I can't believe that [fill in a name] feeds her rat a seed mix, when rat blocks are so much healthier. I'd only ever give ~my~ rat the blocks as a staple diet, so he could be as healthy as possible."
And so on.
OR, tell good stories! "I just met a lady on the rat board, and she never liked rats until her daughter brought one home. NOW, she has ten, and they all know their names, and come when you call them! Isn't that darling? They must be so smart!"
Maybe you could tell some stories about rats on the board, to "show off" their personalities...just in passing conversation, y'know. I tell everyone about Noah the rat. "Can you believe that he won't eat the same flavor jello twice in two days? This is a really spoiled boy, but he's so darling and cuddly."
It's difficult to mount an argument against you, when all you're doing is passing along truthful stories that you've read at the rat forum. It's only making conversation, after all...not arguing.
I encourage you to stay on the forum...there are so many cute stories to share...and photos, too! I posted one here in "General" that is entitled (something like,) "Brave Rat Conquers Feline." It's a picture of a rat who "caught" a kitten. SO cute! You might invite your mom to look at this, and other cute pictures. DO NOT say, "I want a rat," at this time; rather, just use it as an opportunity for the two of you to share and enjoy a darling photo.
The key with this whole approach is not to come on too strong. It's not head-to-head fighting. It's just conversation. Don't let emotions get heated up over rats. Just have it be a casual thing that you mention from time to time. Pretend that she never said anything about you not having rats, and that you are, definitely going to get rats...it's just a question of when. If she takes the hard stance, and says, "I told you, NO RATS," just play it cool, and say, "Oh, I know. I just thought I'd share this with you, since I have friends on the rat board."
And, y'know what? Even if she NEVER comes around to the idea of rats, you WILL still get your rats, someday, so it doesn't hurt to talk as though you're going to have them, because you ARE...someday.
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Someone asked if you know anyone with a pet rat. If you do, maybe you could work something out with them where, when they "go out of town," (wink wink,) you could "babysit" their rats, say, over a long weekend. If you do this a few times, your mom might see for herself how wonderful they are. I've always maintained that the best way to get people to like rats, is to let the rats convince them. (Don't know if this idea will work for you, but I'm trying to think of everything that I can.)
Do you have other pets? At the very least, when you go to the pet store to buy supplies for them, bring Mom along, and walk with her past the rats. Point out how cute and loving they are. While you're at the store, ask someone who works there, "Do you have much problems with the rats biting?" They will of course, respond, "The rats almost never bite, but the hamsters can get a bit nippy."
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This is all I can think of right now, but if you truly want rats, then don't settle for the hamster. Think of this as "the rat project," and slowly, slowly work on getting your rats home.
Hope that something here helps.