I would plant a row of Hostas. They love shade, and would look nice along there.

There's a Hosta forum here-- Check it out

I would bring your front stone wall border around front of this area and plant flowering shrub. The stone on side is perfect for this area. Plants will increase maintenance. Perhaps add white gate to that side yard to give home more polished curb appeal.

Something like this.

"We ... then put in a ditch/drain next to our foundation which we covered in white rock." Please explain the mechanics of this solution. How does water get out of this ditch/drain, and where does it go? It seems to me potentially problematic and that this night not have been the correct solution from the start. Since the space is primarily serving three functions: access to the back yard and for maintenance within the hall, access to light, and for drainage; the simple obvious solution is extending the neighbor's concrete walk to also serve as a flume of sorts, allowing for access and drainage at the surface, without the complication of taking water into an underground system. To be sure, you would not want to invite water into the ground next to the foundation. Plantings might only be necessary near or at the openings of the hall, but are probably not desirable within it on account of potentially restricting drainage, access, or light.

I'm amending to add that the downspout adjacent to your porch should be faced to run into a side walk that if serving as a drainage flume ... instead of being extended with a homely looking corrugated pipe that runs directly toward the front.

And if you wanted to limit access, a decorative gate of some type, even with an arbor appearance, could be added.

Pull out the edging and mix the two colors of rock together.

@Yardvaark Thanks for the information. The PVC drain pipe under the white rock empties into a underground pipe (sewer) that the downspout is no longer attached to (for various reasons). While the corrugated pipe is not the optimal solution, its undesirable appearance has minimal impact in comparison the the state of the rental house next door. @Flo Mangan love the idea! However the sidewalk and backyard pictured belongs to the neighbors house, our property line ends just shy of the sidewalk edge, our backyard access is located on the opposite side of the house. I love the idea of extending the stone wall border.

I wondered where the "property line" was. Any chance the neighbor would be open to putting an arbor up in any case? It would look so sweet and inviting. I would keep the stone as is, because it adds interest without much maintenance. Although, stone does usually require weeding because despite our valiant efforts to keep the little devils out, they fly in and root beautifully in the residue leaf, and dust and dirt that comes into our yards. Of course, a plant won't grow, but a weed will! lol Just extending your bed and stone wall will really be nice, especially if you can put a taller slim evergreen to kind of disguise the walkway.

Wouldn't the pipe running under the gravel be a collector pipe? It would be collecting some water, and some water would be running into the ground below and around it. I'm not saying the black drain pipe is the most horrible thing in the world, just unsightly. My point is merely that neither of these two circumstances are desirable or necessary. And both could have been avoided if a flume/walk was present and the downspout aimed to it. And you wouldn't need to be dealing with rock and how to "save" it.

Thanks @Flo Mangan for the great advice! Would love to extend the bed/wall. Not sure we have time currently but definitely at some point. Thinking about planting a small/medium evergreen shrub at the front of of the rock for now. Any suggestions on what to look for?