A website I'm using on only uses a very small amount of jQuery, in the drop-down menu (I'm using the JSDDM drop-down menu). Is it possible to either manually or automatically make a cut-down version of jQuery which only contains the needed code?
I would simply use the packed version of jQuery http://jqueryjs.googlecode.com/files/jquery-1.3.2.min.js and then compress any additional plugins.
jQuery in itself you can't cut down on. If you use jQuery UI you can build your own dist (http://jqueryui.com/download).
Alternatively (to save load on your own servers) you can use jQuery from Google http://code.google.com/apis/ajaxlibs/
Why do you need this? You can use minified jQuery version (19kb) or write your own JS not using jQuery.
Use the minified version of the library. But you cannot really just remove the part of the library without breaking it.
You could manually strip out what you don't need but I wouldn't envy you trying it, the minified version of jQuery is only 58kb in size tho.
I don't think it's a good idea to edit the code. Use the minified package instead.
Of course it is, jQuery is open source. The source code is here. If you can use only a subset, and probably combine it with your own JS file before minifying, it could be a way to go in terms of load time.
Note veggerby's reply about CDNs like Google Ajax API stuff; since the user will have it in cache if they've visited any other site using jQuery and loading it via Google. Cache is even faster than loading a cut-down version from your site. :-) And Google's CDN is pretty quick, too.