Go back to Tifa's house in Nibelheim with Tifa in the party after Disk 1 and play the song from Cloud's flashback on her piano:
[X], [Square], [Triangle], [R1]/[L1]+[Triangle], [R1]/[L1]+[Square], [X], [Square], [Triangle], [R1]/[L1]+[X], [O], [X], [Square], [X].
Talk to the Weapon Seller, then go talk to the Sleeping Old Man when the last two digits in the number of battles you've fought are the same to get Mythril.
Give it to the Weapon Seller and open the small box upstairs.