How do I map a SQL Server login to a user that already exists in a read-only database?
Since the status of the database is Standby, I'm assuming this database is a Log Shipping secondary. If that's the case, the database cannot be written to at all on the secondary (making it writable would break Log Shipping).

Also, since you said that other databases (presumably not read-only) have correct mappings, I'm assuming you're using a SQL login. If that's the case, all you should have to do is recreate the login on the secondary server using CREATE LOGIN ... WITH SID to match the SID from the primary (look in sys.server_principals on the primary server to find it).

On the other hand, if you're using Active Directory for the login, you'll have to add the user principal to the primary database so that when it comes across to the secondary in a log backup, there's no need to do any remapping (note: depending on the situation, this may be a lot easier said than done!).