Have you thought about trying to just go with it, and making it a marsh/pond? There are lots of plant ideas that will work with a scheme like this. I don't know if you have a homeowners association that would prevent you from making a body of standing water, but if not, that is something I'd consider. I'd love to do something like that with water-loving grasses, irises, etc. Maybe add a little wooden bridge over the boggy area, to give you access to the planting areas beyond?
Also FYI if you don't want stones/gravel to continue to sink into mud you can lay down landscape fabric using staples. It won't stay perfectly in place because of the water issues, but it will contain the stones.
The other idea would be to install a french drain but obviously you'd have to have a channel to send the water into, ie: a destination. In a subdivision that may not really be available to you though, since you can't/don't want to send it out into the street because of erosion.

mini dry well. not sure the volume of standing water you're dealing with but if you excavate say 6' by 3' by 2' deep, line hole with geo textile fabric, fill 4/5 with clear gravel (3/4"), cover with geo textile fabric top up with decorative river rock or other, grade so that water drains to this area. You could play with it; shape it like a dry creek plant decorative grasses behind, a small boulder or two, ground cover in the front... to make it more appealing and looking like an intentional design.

This dry well will essentially act as a natural storage basin taking standing water from the surface where it is a nuisance and a mosquito motel, storing it below the surface from where it will then slowly percolate deeper into the ground. The size of it depends on the amount of standing water you deal with during seasons of heavy rain. Not a small operation necessarily but the only tricky part is disposing of the cut earth you remove from that hole. (Berm up another part of your garden, rent a small dumpster bin?)

good luck

Plants that tolerate standing water, for short periods of time but will also live without it:
Willows
Alder
River birch
Baldcypress
Redtwig dogwood
Sweetspire
Aronia vars.
Many varieties of daylily
Yellow Iris (I. pseudoacorus)
Ajuga varieties
Houttuynia, Chameleon (Houttuynia cordata ÂChameleonÂ). Careful! Can be invasive!
Liriope varieties
The best solution is to get your neighbor to let your drainage go away through their yard, a tough one...taking soil away vs. adding is usually the best answer, honestly! More soil usually dams it up for someone else, just like has probably happened to you.
Hortster

Pictures would be helpful.

I like the wooden bridge idea that will get you from one part of the yard to the other, especially coupled with the idea of a rain garden. Rain gardens help the environment, besides becoming an aesthetically pleasing way to deal with the problem.

Here is a link that might be useful: Rain Gardens

Think Rain garden, plant flowers and grasses that would do well in those conditions.

Some of my favorites include asclepias incanarta, and joepyeweed!