This is what I understand you to have as a relationship:
PasswordTable::Password = RelatedTable::Password
PasswordTable::AccountName = RelatedTable::Accountname
Either way, when you edit the text field in PasswordTable, the connection to the related table is broken. The best solution is to not set this type of relationship up in the first place.
PasswordTable::SerialIDfield = RelatedTable::IDfield
Now, you can edit either AccountName or Password in the Password table and you will not lose the link to the related records. With your current relationship, You have to find all the records in the RelatedTable for a given incorrect password or account name, correct them and then you can modify the key field in the password table to match. While this can be done with a script, failure to modify all the records in exactly the same manner will leave you with missing related records or even related records that match to the wrong PasswordTable record. Thus, my suggestion to avoid this kind of relationship all together.
This was actually set up by a company that the school district I work for contracted to set up. It was in place before I worked here and I have been asked to look into taken over their work since they pay this company for any changes to the database. I am really not to familiar with the program and this is the first roadblock I have run into.
If you are going to replace them, be prepared to invest the needed time to learn how the system works. There are books, tutorials and training courses available that can help. And you are always welcome to ask questions here.
Thanks again, I will also contact the company that currently maintains the database and see if they can provide me with as much info as they can so that I am clear with what I am dealing with. The account I have has full access and the password field in the password table is not modifiable. It seems their is a script that runs once you change the first name or last name or first/last name field the credentials for the password field adjust accordingly. I assume once they have modified one of the editable fields on the record they then reflect those changes on the account in File menu > Define > Accounts & Privilege (filemaker 8.5). On the account the user name reflects the password indicated on the record and the password for the account Is not visible due to it being sercured.
It's hard to say much more without being able to open the file with a full access password and physically examine the file, it's scripts and layouts. There are a number of strategies possible that do this type of thing.