Thanks for posting.
Changing passwords from FileMaker Go is currently not possible. You will need to load the database in FileMaker Pro and use the manage security options there to make changes. The FileMaker Go Development guide at http://www.filemaker.com/support/product/docs/filemaker-go/fmgo_development.pdf goes over password functionality starting on page 10.
I encourage you to enter this suggestion into our Feature Requests web form at http://www.filemaker.com/company/contact/feature_request.html. I could easily copy your post and paste it into the web form, but there are personal questions asked that only you can answer.
After further research, there is one way through FileMaker Go in which a password can be changed. You can setup a script using the change password script step, which could run on opening or off a trigger. Here is an example of the script step format.
Change Password [Old Password: <old password>;New Password: <new password>; No dialog]
The user must have password change privileges for this script step to work. This script would not allow the user to change their access privileges on their login account.