You basically already answered your own question. If you amended the website's name in the Website Maker settings and republished the site now, this will be picked up by Google sooner or later. Search engines basically scan the web on a regular basis and will come back to rescan sites repeatedly to update their own cache. It's hard to predict (for me anyway) when Google will pick up on the changes, but it will.

If you want, you could try the manual submission form for Google:

www.google.com/addurl/

Maybe you can speed up the process or generally improve your indexing/ranking this way. Could always give it a shot...