It's a tough challenge. Unfortunately, the major strip resorts know they have a captive audience, so they don't give you good cheap food options. Your $10 budget is going to get stretched even with a cup of coffee and order of pancakes at a strip resort coffee shop.
For good cheap breakfast, you have to break outside the casino bubble. If you've got a car, there are lots of non-resort places. For typical breakfast fare, see check out The Cracked Egg
http://thecrackedegglv.com/ or Blueberry Hill
http://www.blueberryhillrestaurants.com/
If you have no car, it's a bit tricky in the South Strip neighborhood you are considering. The Showcase mall is in front of the MGM - there's both a food court and a Mexican restaurant (La Sala) that serve decent breakfasts. You aren't going to find a "great" breakfast there, but you can get some decent food for under $10 and won't feel ripped off. There's also a McDonalds in the food court of MGM itself.
Luxor is slightly trickier. My favorite breakfast would be at the House of Blues - an easy walk over to the Mandalay. It'll be a bit more than $10, but much better than what you'll get at the Luxor coffee shop for about the same price.
Now if you want one great breakfast, don't mind going over $10 and have the ability to travel, go to Bouchon at the Venetian
http://frenchlaundry.com/bouchonLV/bouchonLV.htm/ It's a beautiful place to eat, and the food is to die for. Try the French toast - I promise you'll come back here and thank me. (One note - Bouchon is actually kind of hard to find at the Venetian if you don't know where to look. It's not by the other Venetian restaurants, or the Canal Shops, it's in the Venezia Tower. But it's very easy access from the valet/self-parking once you know where you are going.)
Thanks! 
