Outlook losing signatures

Since it is only happening to a single user, I believe the MAPI Profile is corrupt.
Have you tried removing and re-creating the Outlook profile?
Right-click the Start menu and open the Control Panel.
Select User Accounts, followed by Mail.
From the Mail Setup window, select the option to Show Profiles.
Click on the profile you want to delete and choose Remove.
Click Yes to confirm in the pop-up window and when it's done, click OK to finalize.

Then create a new profile:
From the Control Panel, choose User Accounts, followed by Mail.
Inside the Mail Setup window, choose Show Profiles.
From the Show Profiles screen, click the option to Add a profile.
Name the new profile and click OK.
Follow the prompts to create the new profile.

If you keep losing the signature even after creating a new profile, then I would run an Office Repair, and if an Office repair fails, then try to see if the problem happens on a separate computer. If the same user account does not have the issue on a separate PC then we know the problem is isolated to that user profile. Removing the user profile and re-creating the user profile is a pretty big step just for a lost email signature but it's the last resort.

To run an Office Repair
Open Control Panel (Start>Settings>Control Panel)
Select Programs and Features
Select Microsoft Office
Click the Change.
Select the Reinstall or Repair button, click Next.
Select Detect and Repair.
Click Install and follow any prompts provided by the repair tool.