I am making some changes to a website for a friend, (because the guy who created her website charges $50 an hour, and I'm only $25). And I noticed that he has her set up with an index.html, and a home.html. Both of the pages are identical. When you type in just the url, it takes you to home.html. And the links on the top of the page to go to the home page all point to home.html. So what is the point of even having an index.html? Is it going to be bad if I just delete it? Will some browsers point to the index.html? If so, that means every time I make a change to home.html, I have to make those changes to index.html as well!

Typically, in the absence of a particular file in a URL (the URL points to a folder), the web server is configured to deliver an "index" file inside the folder. Traditionally, this is index.html, or index.php, etc. It doesn't have to be. It sounds like the developer started with index.html, and for some reason later created home.html, set the web server to deliver it, and never deleted index.html.

In other words, you don't have to worry about browsers: they don't determine what file gets served.

Thanks, so it sounds like I could delete the index.html, and it would be just fine? I just don't want to have to keep worrying about updating it.

Yes, that's what it sounds like. Be sure to make a backup and test the site, of course.