This page of our game guide to Control - AWE contains a detailed description of the Gerbil Took the Top Head side mission. By reading our walkthrough you can find out how to select the right gift for the prisoner, where to find the helmet and how to obtain it.
How to unlock the task: This mission will be unlocked automatically after driving Hartman away from the Fra Mauro AWE location.
On the ground floor of the room, where you had to dodge the boss, there is a Holding Cell as shown in the picture. Talk to the person being held here.
You must first determine what kind of item the prisoner is interested in. You can't find the right gift right now. Explore the nearby labs - Mobile Lab 1 and Mobile Lab 2. Bring any items which you can find in them and put them in the container. Do this until the mission objective gets updated.
The proper gift is a helmet and in order to get it you have to go back to the Sector Head Office, which is the first location you visited when you first got to the Investigations Sector (there is a checkpoint near the office - use a fast travel option).
Be careful, because the access to the office is guarded by an elite enemy who has spawned in this place since your last visit. When fighting an elite Sergeant Ranger, help yourself by using a shield and perform some sideway dodges and evasive movements. You can also hide and take cover behind poles and walls.
After winning the duel, explore the south office upstairs - you'll find the helmet shown in the picture. Return with him to the prison. Place the helmet in a container located in the cell in order to complete the quest.