You know what? I have one of those!!
Just like you only narrower. In the flooded areas nothing will grow except for weedy grass and moss. I have tried, and tried but fail every time.
We just had some french drains put in, in some new spaces we created with similar problems, after a renovation. I highly recommend you do the work to put in a french drain system. It will stop the mushy flooding, and then you can put in a narrow walkway and have more planting room!
Instead of rock paths, how about just using a mulch path? Shredded hardwood mulch from a tree man is free and it works well in a shaded location like yours. Or you could put in your french drain system under a poured concrete path. My new areas have poured concrete paths 3' wide from the driveway into the back yard because it really helps with access for large deliveries, wheelbarrows and if you ever need to have a flat path for walking (like I did when I recently broke a foot).

Just some things to think about!

GGG

Hey GGG. Funny you mention the french drain. If you look towards the back right of the photo, you'll see some pipe rolled up in the corner behind the wheelbarrow. That's the leftover from the drains we put in the backyard and behind a retaining wall. The problem with doing it on this side of the house is that there's no good place to drain it. The natural slope of the land runs towards the front yard, which would create a run off disaster in that area.

I've thought of digging a pond for it to run into, but DH does NOT like that idea. I also thought about a rain garden, but DH has opposed that too. Guess I just need to keep wearing him down. hahaha.

Mulch is a great idea, and would be fantastically simple (and cheap - which I love). However, in a storm, it will lift up and float away. This area has been my nemesis since we moved in, and that's probably why it's never been finished. :-)

DH is running out of options, isn't he? The french drains can be buried to run right out into the street. You can also make a drain pit. I don't see a driveway there yet? If you don't have one, you could make a drain pit under the driveway. We did this under a planting area (where french drains are) although there is also one pipe that goes into a "proper" drainage area. I think what is happening is that you get water collecting and then you get all the water from your downspouts that are tied into the ground. You have to move that water away from your foundation. After you put all your drainage in you could put brick or pavers right on top of a drainage pit and make a walkway with them. There is a home on Woodland Ave. near me that has done this, it is so classy looking.

GGG

it would make a great Japanese garden with lots of gravel and rocks and a couple of plantings. That should not wash away and the area could be made to be very attractive and calming (just what you need after trying to decide what to do with it!)

It looks fairly well sheltered from winter winds. Have you considered planting some small flowering trees (dogwood, redbud, silverbell, witch hazel, japanese magnolia, japanese maple or crape myrtle, for example)for summer shade? Once they have achieved a little size you could underplant with daisy gardenia, small azaleas, pieris, korean spice viburnum and hydrangeas. Some camellias espaliered against the fence would be beautiful. You could have a really pretty little wandering path down the center (with only eight feet of width it would have too :), maybe a small patio off of those doors?

Constrained spaces, I find, are often more fun to play with than expansive ones.

A good mulching does wonders for poor soils much faster than you might think! Hardwood mulches can absorb massive amounts of rainwater. And there is no need for stepping stones.