You'll pick up this quest in Chapter 14 after hearing Kyrie's speech in the Sector 5 Slums. Make your way over to Wall Market and as you head inside, you'll spot Betty from Sector 7 walking towards the courtyard to the east of the Hotel.
Head on over there to speak to her and you'll discover that she's looking for some Music Discs to help lift the spirits of the people around her. You can get these in any order and for each one, you must play it in the Jukebox that she's standing at for it to register.
The first Music Disc, Good Night, Until Tomorrow, can be found nearby in the Hotel, head inside and speak to the man in the corner to receive it.
The second Music Disc, Stand Up, can be found by heading to the Honeybee Inn and then heading to the alley in the northwest corner. You'll find a woman here listening to the track who will hand an extra over to you.
The final Music Disc, Fight On!, can be found at the Colosseum Souvenir Shop for a paltry 50 Gil.
Once you've played all three on the Jukebox near Betty, you'll complete the Quest and obtain the Shapshooter's Companion Vol. III which will increase Barret's SP by 10.