As we all know, the URL has some importance in SEO.
Should I try to rewrite (or redirect) the index page to a specific-name?
index.php to white-cats-are-cute.php
(This is regardless of the domain name or the directory name, it is very specific to the index file)
The case is a bit specific. The domain itself has no keyword, but an abstract name. Let us imagine something like example.com which says nothing about the type of product or services.
It is a One Page website, so it has no other pages that could hint about the product, like example.com/hamburger.html, so the only page it has is index.html.
So, yes it is to have a keyword on the URL, (of course besides a good title and description)