I was listening to a podcast yesterday and they mentioned that in Adventures in Middle Earth there's a mechanic for levelling up magic items so they get stronger or add special abilities as the character gets stronger. They didn't go into detail about how it works but it sounds like a cool mechanic and I was wondering if anyone has done something similar?

Yes, there are a few systems for this>

WoTC did Weapons of Legacy, and then revisited such in 5e with the Storm King's upgradable (slotted) items.

Here is a homebrewed upgradable magical item system with some examples implementations:
http://homebrewery.naturalcrit.com/share/ryXHUftFZ

Also here is another member's version of such

http://www.dndbeyond.com/forums/dungeons-dragons-discussion/homebrew-house-rules/10386-looking-for-feedback-on-artifact-weapon-that


I'd not use that personally. but you can see the nature of another's efforts down that rabbit hole.