On this page of the game guide to Borderlands 2 you will find a detailed description of the side mission titled The Name Game. This is one of the few side missions that you can begin during your first visit to the Sanctuary.
You can receive this mission during your first visit to Sanctuary.
1) Moxxi's place
Three Horns Divide - northern part
2) Bullymong lair
Three Horns Divide - southern part
3) Bullymong lair
4) Bullymong lair
Head to Moxxi's  and speak with Hammerlock to receive this mission. Head to Three Horns Divide and drive to one of three Bullymong lairs , , . You need to kill 15 bests and search 5 piles shown in the above screenshot, which can be found near the lairs.
If you complete this errand, Hammerlock will tell you to kill a Bullymong with a grenade. Your next task will be shooting down three rocks thrown by Bullymongs, in mid-air. Keep the enemies at distance to provoke them to throw rocks. With three rocks shot down, you will still need to kill a couple more Bullymongs to finish the mission.