Where are you? This is important since it will influence what plants you can use as well as other factors such as need for snow removal. Here's how to add that information to your profile so that it posts with your name:

Go to Your Houzz in upper right of every page, click Edit Profile, and on the left side click advanced settings. Well down the advanced settings page is a blank labeled Climate Zone for Garden Forums along with a link to find your zone.

If you add info on your state or the nearest large city you will get even better information since zone only relates to average coldest winter temperatures.

Then return to the top of the page and click Done Editing.

I am in the northern part of South Carolina, on the edge of the mountains. My zone would be 7a. I was told I don't need to worry about snow.

Other parts of the sloped driveway have a thick ground cover vine is that is working well.

This part of the driveway has morning sun filtered by trees, but the remaining day is full sun.

So I was hoping to plant some rabbit eye blueberry bushes

Blueberry bushes would be nice, but they are not evergreen and will not stay low and be successful ground cover. The rabbit eye can grow 15 feet tall and 10 feet wide. You could just not prune the bottoms and trim the tops I suppose. Do you have deer to worry over? They will love your addition. If this were me, I would wait before I did much change, river rock might be a lot of work if the slope is one that runs. Mulch is much easier to sweep back up. What is the thick ground cover vine that works well? Get more of that.

I put in a narrow metal edging in a neutral color in a situation with an even smaller grade. It doesn't attract attention.