The widget removes the text from the title
attribute to prevent the native browser tooltip from appearing. When it removes the text, it stores it in data attached to the element.
You can do the same using this line of code:
$("#my-element").data("ui-tooltip-title", "My new tooltip message");
Now, if the user moves their mouse away from the element, and then hovers back onto it, it will show the new text.
To show the new text immediately, you need to update the live tooltip overlay element, which thankfully is easy to find. You just need to do this after you've updated the text:
$(".ui-tooltip-content").html($("#my-element").data("ui-tooltip-title"));
Tested on jQuery UI 1.10.0.