FileMaker Go is not a fully-functional FileMaker client. Many of the features implemented in FileMaker Pro and FileMaker Pro Advanced are not available in Go - including pretty much anything having to do with design or development, and also including the Security dialog.
However, if you put the database under Pro or Advanced, you can indeed write a script that will allow the user to change a password. Look under the Accounts section for the relevant script steps.
It works I created a button and attach the "Change password" script to this button.