Greasemonkey is certainly an option. Another idea is to code up your improvements, and then make bookmarklets out of them. Your friend can use the bookmarks (probably in a bookmark bar) to do the things you've improved. Bookmarklets have access to the page as though they were a part of the page.
Edit 1 In fact, now I think about it, a bookmarklet should be able to load a script file (from a different origin) into a document by adding a script
tag to the head
section (well, or anywhere, really). Since the SOP is based on where the document came from, not the script, ... That way, he'd just have to press the button once (for any given page he goes to) to load your improvements.
Edit 2 Yup, a bookmarklet can be used to bootstrap any script file into the page; here's an example:
javascript:(function(){var%20d=document,db=d.body||d.documentElement,elm;elm=d.createElement('script');elm.src="http://example.com/yourscript.js";db.appendChild(elm);db.removeChild(elm);})();
That adds a script
element for the file http://example.com/yourscript.js
to the body of the current document, which executes it. (The bookmarklet then removes the script
element; just adding it is enough, it doesn't have to stick around; details.) Your script can then do things like add other scripts (jQuery, in your example) in the same sort of way, fire up auto-completers, etc. Tested the above (which probably needs tuning) with Chrome and Firefox; IE isn't liking it but I think that's an issue with my encoding of the bookmarklet or something rather than a fundamental problem. (I'm relatively new to bookmarklets.)