Ideas on what to do with that skinny strip. It's about 1 brick wide.

I would try an herb garden there. Get some garden dirt and plant herbs or tomatoes etc. Cucumbers would grow great against the fence.

Plant some pretty annuals. Or perennials if you want less work every year. Keep them small with soft colors. I saw a house that had done that and it was so pretty.

amend the soil and plant succulents or cactus, or use small drought tolerant plants

You can plant annuals or perennial dryland plants like sedum, hen and chicks, violets, etc.

I’d use either mulch or small stone. Both will provide good drainage but stone will last longer.

With a chain link fence.... stone or mulch would be escaping into the drive way all the time with out a border strip.... that area would be a good place to plant some of the lower growing ornamental grasses... low maintenance and they can grow in not great soil

If you don't want plants in there, I would buy small colorful pebbles and fill the space. Can't tell how big it is so cost would depend on size.