A Better Gravity Well - Hammerhead
First Available: Chapter 5
Other Requirements: Acquire Gravity Well
Rewards: 2,000 EXP & Gravity Well Plus
You'll be able to accept this quest after obtaining the Gravity Well item which is found in Aracheole Stronghold, an Imperial Base that you visit during Chapter 5. The Gravity Well item is found in a hangar in the center of the base, as shown in the screen shots below.
For upgrading this weapon you'll need to bring Cid a Hydraulic Cylinder which can be found as a spawn in many different locations throughout the game as well as dropped from an MA-X Patria which is a Mech enemy that can sometimes appear from the Imperial drop ships that follow you around in the Leide region.