I just tested this but I can't replicate it. I'm able to create a custom field, set a default value (???) and then change it on a per-ticket basis as an agent and the default value stays the same. So, let me ask a few questions:
1. What does the custom field apply to? People or Data?
2. What's the Data Type of the field?
3. Does this happen with every custom field or just one?
4. What Role do the "users" who change the custom filed have? And are these Agents who are changing the field or actual end users?
If you want to screenshot the settings for at least this Nickname field and DM them to me, I'd be happy to take a look.