You are looking to get rid of surface water. The drainage description that you have is either to remove water from saturated soil or to disperse water that has been diverted to the pipe to have it leach into the soil. You need to get that surface water into the pipe first in order to make that work oer to expose the drain rock on the surface (don't cover it with soil or grass).

You are missing the point or area of entry for that water to enter the drainage stone. One choice is to put in a catch basin for the water to enter the system at the lowest point of the swale. It would be important to put a silt fabric over the stone and then at least 6" of soil for your lawn to survive over the stone.

Another option is to make a trench that is filled all the way to the surface with drain rock. You would be looking at the drain rock, so you would probably like to make it look more interesting like a dry creek bed. You would not need a pipe with that method.

Check out Home Depot or Lowes, as they have pre-fab dry-wells and how-to books on french drains.

I think you need to provide more information to know if this is a viable solution. Do you know if your subsoil conditions include a less permeable layer over a more permeable layer, which may be causing the slow percolation? If the soil is uniform in texture for the top 4 feet or so, a dry sump is not likely to make much of a difference.

Also, how much water ponds at a time(a couple of inches, a foot, etc, and over how large an area?), and how long does it take to disappear in a typical rainfall event. Under heavy rainfall situations and without an impermeable layer to be broken through to a more permeable soil layer below, a dry sump is unlikely to make any difference in high rainfall events, unless it is sized sufficiently to handle all the water.

Might it make more sense to treat this area as a boggy planting area and lose the lawn, and plant water loving plants?

Sounds like you are good candidate for a rain garden.

Good advice so far. If you are trying to get rid of a small amount of water you can try this. Dig a big hole. Fill it with pails set up side down. The pails need some holes in the bottom (which is now the top) to let air out. cover with rocks, sand etc. this provides much more space for water than filling the hole with rocks.

A rain garden or bog is an even better idea.

In the dry well coil the perforated drain pipe to the bottom rather than just having it hang into the top of the pit. This will prevent the exit area from the pipe to the well to become clogged with sediment and less effective over time and will also increase the holding area of your well by having less crushed stone. A little more cost in length of perforated drain pipe (roughly 100$ for a full role) but worth it.

We had a similar problem at our previous home. There was a drain outside in the entrance way to our basement. Every time we had a lot of rain, that drain would overflow and after a while starting coming into our laundry room. I was talking to somebody and he said that was probably no more than a 3 lb. coffee can inside that drain, and on top of that there was a swail that ran between our house and the next door neighbor, taking the run off of rain water off townhouses built next door to that neighbor. Water would stand in that swail so we dug a trench all the length of that swail on our property, and laid down that black corregated pipe with holes in it and about every 10-12 ft. we built a box frame and found a business that sold cast iron grates that size. We repeated the frame and the pipe all the way to the public sidewalk, took a post hole digger and dug under the sidewalk to the grassy area between the sidewalk and curb. Continued our drench, of black pipe, broke up a small piece of curb so the water could flow out of the black pipe and patched the curb, so the water ran off when it rained hard into the street and down into sewer nearby. We installed a small sump pump in that basement entrance way and it was hooked up to an inside electric plug that would pump that water up another pipe and out into that drain field we'd installed over that swail area. A lot of back breaking labor in 95 degree heat but hey, it worked and still exist today. I am sure a professional company would have charged us a fortune to do what we did. Maybe they would have done it a different and more efficient way but we didn't have that kind of money to spend.

In answer to your question. A dry well is relatively easy to build. Assuming you are trying to proved more volume to allow water to drain faster into relatively porous subsoil: Dig a hole, 4 or 5 feet in diameter as deep as you feel needed, probably 3 or 4 feet. You will need some reasonable rocks, the larger and more easily stacked the better. Lay the rocks so they nest together in a stable manner while still allowing voids for drainage. Fill around the outside with 3/4" crushed stone and place about a foot in the bottom. Find one or two big flat stones with which you can cover the top SAFELY. Cover the top stones with hay or upside down sod to keep the soil from washing in and filling up your dry well/making a hole in the lawn until the new sod establishes it's root system. Use the soil from the hole to re-grade the majority of the hollow out of that point in the lawn. Re-seed. As for pumps, unless you have a place on your land to put the water, many communities have laws against passing your water problems on to the next fellow. A dry well overcomes this issue.

I forgot to point out, you are building a "dry well". Do not fill the center with rocks. Leave a 2 foot diameter void for the water. Hence the need for one or two "cap stones".