Ah. I totally know how you feel. Believe me, I've had too much experience with things like this.
Start researching on eBay for the one you want. Keep an eye out for some nicely priced ones, as you want to spend as little money as possible in order to increase your chances of winning your mom over.
Once you have found the one that you want and is at a good price, sit down with your mom and explain what this is, how you will pay for it, why you want it, and emphasize that this is something you want to spend your money on and that it will be worthwhile. It is also a good idea to tell her that this won't get in the way of school (which it won't, because the iD L is very low maintenance). Finish by asking her very politely if it is okay that you order it. Patiently answer any questions she asks.
The most important part is: if she says no, don't whine or complain. That's really immature and will actually decrease your chances of getting one sooner. Just thank her for her time and leave.
If, on the off chance she does say no, be patient! Wait until when Christmas or your birthday roll around, and start leaving little eBay links around all the time to the one you want (I did this all the time xD). There is a more likely chance she will get it for you.
I wouldn't recommend lying about anything, because if she finds out, that will decrease your chances of getting one, or ordering one behind her back, because that will really show to her that you are being irresponsible. :/
Good luck with getting an iD L!
EDIT:
THIS and
THIS are the two cheapest options on eBay currently. I would wait a bit for a cheaper one, but that's just me.