After further investigation, I think I've discovered the root of the problem:
It appears that disabling the browser's native autocomplete functionality for an input also disables its ability to remember what its value was when you return to the page via the back button. This appears to be true in both Chrome and Firefox. Here's a simple test (runnable at http://jsfiddle.net/TylerRick/aaRSk/):
<input autocomplete="off" type="text" name="test" value="change me"/>
To reproduce behavior: Change the value of the input, then click the jsFiddle logo in the top-left corner, and press Back. It will have reverted to its starting value of "change me".
This is the line in jquery-ui-1.8.16/ui/jquery.ui.autocomplete.js that disable's the browser's native autocomplete functionality:
.attr( "autocomplete", "off" )
When I remove that line and try leaving the page and pressing Back, it does in fact keep the value as I left it just before leaving the page. And the jQuery UI Autocomplete plugin is still fully functional.
However, the browser's native autocomplete drop-down list drops down ''over top of'' the plugin's drop-down list, which is probably the reason the author of this plugin disabled native autocomplete.
The native autocomplete can apparently be dismissed with Esc without dismissing the plugin's list (a second press of Esc will dismiss that list too), but leaving native autocomplete on is probably still too much of a usability problem to recommend removing this line.
Having said that, does anyone know of any other solutions to this problem?