Back up with camera to show surroundings and beyond.

I like the river rock idea. That's what I use in my flower beds on drainage channels. I will dig them out more and the lay down river rock from Lowes or Home Depot.

I'm not sure if I understand your pictures. In the bed where the roses are, is that at the end of your driveway? In the picture of roses, the basketball backstop that shows up is the same one ... so it is in your driveway, not the neighbor's yard? The water that is being dumped on the roses area is coming from your driveway? If that's the case, we need to see the last 10' of your drive x 20' of its width ... around where the covered car is parked. About from the trash cans to the back of where the car is parked. You may need to move the car. It is fine to capture in two slightly overlapping pictures taken from the same camera position so you don't have to get too far away from the subject. You might include another picture (not a close-up) showing right where the water comes off of the drive.

The solution is going to be in dealing with the water BEFORE is careens down the hill.

Yes the roses are behind the fence in-between the basketball goal and trashcans. That seems to be the low spot in the driveway. Maybe these photos are more detailed. The arrow shows where the water concentrates and exits to the backyard.

It is picture 2 and 3 of this batch that really explain the problem.

I don't know how it would have come to be, but your driveway's end is sitting considerably above grade of the surrounding soil. This is not good because it is subject to (very likely will happen) undermining of the drive. Running water tends to cling to a surface. It will come off of the drive, run down the exposed edge of the drive, and then start working its way back under the drive ... very much like how earth is carved out from behind a waterfall. At the driveway' end, soil should meet the top of the driveway such that the drive is flush with the adjacent grade. The slope that is beyond the drive should be build back up to the drive surface. For the sake of safety and some "cushion", the slope should not begin immediately at the drive's end. It should be level for a foot or two and then let the slope begin.

Naturally, if there is a lot of water coming off of the drive, it's still going to erode a channel in the slope. The solution depends on exactly how much water is coming off of the drive during a rain, which we can't see yet. Suppose it is moderate. If you established a solid, dense groundcover that would probably take care of it. The above-ground portion of the plants lays down when washed over with water and protects the soil. Dense roots hold the plants in place. Roses would not act as a groundcover.

If the water is high volume, you're likely going to need to catch it at the top and convey it to the bottom via either a pipe of a flume. (Still, this does not negate the need to extend the slope by adding more soil in the first place. To have the driveway end sitting above surrounding grade makes it subject to cracking over a period of time.) It's probably more common for folks to use drains and pipes because people like things hidden out of view. My own preference here, though, would be a flume because it would be easy to conceal with plantings and, easier to maintain (keep from being clogges.) And if it were to clog, excess water would just spill over its edges until it was cleaned out. A buried pipe may stop working entirely until repaired. You might think of a flume as a sidewalk with raised edges ... a chute. It's just a path to guide water without destroying anything along the way. It could be built out of concrete, or it could be built out of stones, like a cobblestone walk ... so long as the stones were large and deeply embedded enough. Any kind of gravel or small stones setting on the grade wouldn't work as they would also be subject to washing down the slope.

If you didn't want to extend the slope at the end of the drive, you could install a retaining wall to support the drive. But that would be more expense and effort.

Would the flume run parallel to the fence? Where would it empty at? The neighbors fence is to the right.

No. It is a conveyance that runs down to the bottom of the hill where it is released onto the lawn (which is a thick 'groundcover'.) The picture does not purport to show every detail, but is to give you the general idea. You'd need to catch and funnel the water into the flume at the bottom of the drive. And having the flume does not absolve you of the need to fix/extend the hill. As you can see, a flume would not be likely to clog. But if it did, it would be obvious, and easy to clean.