Hi Kenneth,
I can think of a couple different ways you can do this, depending on the level of customization you are comfortable working with. The first option would be to change the "Invoice Notification" email template located on the Home->Email Templates Screen. You can simply add the main Contacts_Portal link located at Customize->Portals, and select the "Get sign-in web address..." link. Copy the link provided in this pop-up window and paste it as a hyperlink in the Invoice Notification email, check the screenshot below for an example. Keep in mind that this option would require your customer to create a login, and use a username and password each time in the future.
The 2nd option would be to customize the "Save & Email" button located on the "Quickbooks_Invoice" Screen, in the Footer section. This would be fairly simple and only involves modifying one action. However the benefit with this option over the other is that it will do exactly what you are asking, take your customer to the "Home" tab of the Contacts_Portal screen, but keep the same login type as originally available, using just their email address. Go to edit the "Save & Email" button, locate and edit the action with the description "Get resultURLforPortal", then change the line "Tab Link to open" to "Contacts Portal>>Home". Save, close and publish the updates you made to this action and the screen and you should now be setup correctly. Refer to the screenshot below for this final step:
Please let us know if this helps.
-Ben