Prevent loss of unsaved customizer settings or provide means of restoring

[westonruter]

This is to revisit #20692, opened by koopersmith for 3.4:

Losing your changes is no fun. ¶ The way I see it, we have two options:

1) Show an AYS when leaving the customizer by using an onbeforeunload handler.
2) Store unsaved customizations in a customize_data_$stylesheet option. The fact that certain customize settings persist across theme changes (options) while others don't (theme_mods and some theme-specific options) could pose a problem.

In the resolution of that ticket, neither of these options were chosen. The beforeunload handler with confirmation dialog (first option) was discouraged, and then another suggestion was provided to just make the Save button more prominent so that users would not neglect to press it before leaving the customizer. However, it seems the more prominent placement of the button is not enough.

For the Widgets UI Refresh feature-as-plugin effort, a user test was put together to test integration with the Customizer. In the video at ~7:30, after the user had made changes to the controls and previewed them, he navigated away from the page. He forgot to hit Save & Publish, or didn't notice it there, and so he lost all of his changes. See the video here: ​​http://make.wordpress.org/ui/2013/09/18/widgets-sept-16-chat-notes/#comment-23907

For the 2nd option (storing unsaved settings), another solution than storing drafted settings in a DB option is this: now that autosave will store the drafted post in localStorage, we could do the same for settings in the customizer. Since all of the settings are stored in memory as JS objects, when leaving the customizer with unsaved changes it could store those settings in localStorage.

After having accidentally abandoned changes, when returning to the customizer, the initially-disabled "Saved" button could then swap out with an enabled "Restore" button. As soon as a change is made to a setting on the page, the Restore button would replace with the "Save & Publish" button, and the localStorage-preserved settings would be discarded. But if the user hits "Restore" then the localStorage settings would be populated into the customizer settings and the button would also change out from "Restore" to "Save & Publish".