The iMac in question is probably a few years old, but hopefully not too old. So, here are ten ideas for convincing your mother to let you have that iMac (helpfully avoiding all reference to teen-aged obsessions with Facebook, etc):
1. You can get a recent, free version of XCode onto the thing and tinker with programming for OSX, which is a bridge into programming for Unix generally. Since Unix is what runs the web and much of the world, thar be jobs there. Also very useful in science, business, government, industry....
2. You can get a recent, free version of XCode onto the thing and quite possibly actually learn to program for iOS. Maybe even for iOS7 if the iMac is new enough.
3. Registration as a developer is free (and needed for the XCode download), so it will not cost you or her a bean.
4. Sign up for Code Academy (free) and let her see you working at it.
5. It seems not to take too, too much stratospheric know-how to get a basic iOS app published. You might make a few quid.
6. In the (probable) event that you made no money, you could still use the fact that you had done it as fodder for your applications for uni. If you wanted to study medicine (say), rather than Computer Science, it would still help you stand out from the crowd — which is precisely what you want to do if you apply for highly competitive places at uni. Admissions officers are interested in more than just grades. Harness that geekiness and use it constructively!
7. Get her to think that a household iMac will channel some of the energies recently spent on the little LC (something that she frankly knows will never again be useful in the real world) into something that might actually come in useful in the real world.
8. Remind her (politics permitting) of how Bill Gates advised Blair to enable more UK schools to teach kids programming.
9. Promise her that you will clean up your room and eliminate the "floordrobe" situation in future. Bribery is always a solid plan where recalcitrant parents are concerned. Create a "floordrobe" crisis to advance this devious plan if necessary.
10. Get your father on your side. Mind you, it looks like he already is on your side. I think you are probably set.