The path field can't be changed, but it is possible to make the folder tab show the site name, using a script.
The Tab-Labelizer script already exists to do similar things, so it probably makes sense to work with that rather than re-invent things. I've just posted in the thread about it with some ideas.
At the moment you can get something decent, but not perfect, with the existing script just by editing its configuration:
My post in the script thread has the config I used to get that. If it looks like it'd be useful, shout and I can explain in more detail how to set that up. Customizing the script is also possible, or maybe TBone, who made the script, will add some options if you ask (up to him, of course). This assumes you're happy with the details being in the tab rather than the path field; if not then the script won't help.