When I assign a ticket to one specific user, Jira puts the user name in the Assignee field but highlights it in blue. If I tab off the field, it reverts to "Unassigned" (or the last user that I was able to assign).
This seems to be impacting only one particular user.
Navigate to Project Settings for the project in question.
In the navigation pane then select Permissions.
Under the Issue Permissions section you'll find Assignable User. Review the User/Groups/Project Roles that are assigned to that permission, and then confirm the user belongs to one of those.
Also, is your user able to view the issues in the project?