Much of this information was gathered from http://www.petri.co.il/enable_password_changing_through_owa_in_exchange_2003.htm.  Credit goes to its author.

1.  Install the OWAAdmin tool and enable the Change Password feature.
http://www.microsoft.com/downloads/details.aspx?familyid=4BBE7065-A04E-43CA-8220-859212411E10&displaylang=en

2.  Enable SSL on the server.
3.  Create a virtual directory under the Default Web site.  Call it IISAdmPwd and point it to %Systemroot%\System32\Inetsrv\Iisadmpwd
4.  Grant the following rights on the virtual directory: Read, Run Scripts and Execute 

5.  Open the OWAAdmin tool from https://<servername>/owaadmin.
6.  Go to the Security Settings page, and then enable the change password option.

Verify that the Change Password option is available in OWA:
1.  Open OWA in a Web browser using https://<servername>/exchange
2.  Logon with your username and password. 
3.  Click on the Options button. 
4.  In the Options window scroll all the way down. Notice the Change Password button. 
5.  Click it.
6.  Notice the Change Password window opening. Try to use it.