The homepages on my sites have been influenced by what my visitors do when they land.
When each of my sites started, I didn't have much content on the site so my homepage had very few options. Then as the sites grew I added more content options to the homepage and the persistent navigation across the top of the site and down the side became filled with links to category pages.
As I added more options to the homepage user engagement increased, and as I added still more options user engagement increased again, and again, and again. Pretty soon, other people who have websites and strong opinions about homepages tell me I have way too much stuff on my homepage. I can listen to them or I can listen to my visitors. I picked the visitors.
So, what are your visitors telling you about your homepage? Are you doing anything to learn how they behave when they land? Crazyegg gives you the ability to see what they click. It is easy to find out what they are clicking, what they are ignoring, what returning visitors do, what new visitors do... you can even see what visitors on mobile do and what visitors on tablets or desktops do. If you don't know what the visitor is doing you are just shooting into the dark.
You can use visitor information to move heavily engaged items to prominent positions, you can see what isn't being engaged that you want to be engaged and give it different wording or a better photo to see if that improves. It is all about giving the visitor what he/she wants and enticing them towards what you want them to accomplish.
What's all of this have to do with your rankings? A lot, I believe.
If you are giving the visitor what he wants that visitor is more likely to engage your site instead of bouncing, share your site, bookmark it, return to it using your domain as a navigational query or typing it into the address bar of the Chrome browser, click on it preferentially in the SERPs. So, I think that by pleasing the visitor your site will produce signals that google is watching and that will help it rise in the SERPs as long as you have the basic page optimization and a decent navigational structure.
As for what to show the visitor on your homepage, you can send the message that "we ain't got much" or "you gotta click to find out" or "we have an enormous variety of things". I have bet ALL of my money on the enormous variety because that has been what generated the most visitor activity. Maybe your visitors are different from mine, just saying what I believe works on the small number of sites that receive all of my attention.