I think the most influential tool google has made for education is Google Scholar. It has changed the way people can search for academic articles to use in research online. Not every educational institution will have access to databases of articles, which are usually quite expensive. Google Scholar is able to find content relevant to what you are searching for, which can be limited to just articles that are not behind paywalls. However if you are at a university or a public library, google scholar can be built to work with the databases you can access. Instead of having to look through all the databases one at a time you can just search in google scholar and have access to all of them at once.
As far as what Google's most innovative desgin is, I would have to go with Google Docs. As far as I remember they were the first one to create word processing, graph editing, and picture editing all completely online. The innovate part of google docs is how easy it is to use. Anyone with an internet connection now has access to word processing software. Since it is web based, it have very low system requirements, which basically means if you have internet you have a full useable version of Office completely for free. I could see that being used to provide tools to people in disadvantaged circumstances. The best part is that you only need to connect to the internet once. Google docs is
useable offline as long as you have chrome and initially visited the site.