My customer asked if they can change the tool tip from Qucik Search to Page search. I could do that by editing the webtoolbar.html file in the results of their build. I am wondering if there is a way to do this change through the UI of Help Studio so that is stays and does not default back to Quick Search with the next build of the project for that customer.

The webtoolbar.html file is part of the HelpStudio template. I would suggest creating a custom template, modifying the page in the custom template and then you can switch your project to using that Template.