Custom Values and Contact Custom Fields have always worked as destinations for Header Buttons, Contact Buttons & Menu Items.
A heavily requested upgrade to this has been to make it so all the merge tags also work in the button names & menu item names. This has finally now been released.
Merge tags that work:
Custom Values — {{ custom_values.your_value }}
Contact Fields — {{ contact.first_name }}, {{ contact.your_value }}, etc.
Location Data — {{ location.id }}, {{ location.name }}, etc.
User Merge Fields — {{user.first_name}}, {{user.last_name}}, {{user.email}}, etc.
Full list of HighLevel Merge Fields available here. Most should work. If one doesn't, send in a support ticket.
