The sleeping man will give you something when you have won a certain number of battles. You can find him in a cave a little ways south and east of Midgar. You’ll need the Buggy, the Airship, or the Green, Blue, Black, or Gold Chocobo to get to him. If you want to get to him using the Buggy, drive it into Costa del Sol and you will have the option of transporting it back to the continent where Midgar is. Inside the cave is the sleeping man, of course.
He’ll usually tell you how many wins you’ve gotten so far. – If your last two digits are odd, he’ll give you Mythril. – If your last two digits are even, he’ll give you an item (usually a Bolt Ring or Ice Ring). If you have a piece of Mythril, you can take it to the blacksmith who lives in the house on the peninsula near the Gold Saucer. He’ll let you trade the Mythril for a Gold Armlet (in the long wooden chest), or the Great Gospel Manual (in the metal lid on the upper floor). You can return to the sleeping man and get as many rings or pieces of Mythril as you like.