The steps you must take to seed a muddy lawn are no different from those used to seed a dry lawn save that the lawn already has enough moisture in it to help the seeds grow. In fact, a muddy lawn is an ideal environment for seeding as the soil is already worked up and wet enough for germination to take place. Just take care to wear protective clothing and boots while you work.
Prepare the Ground
Test the pH level of the soil. Scoop some of that mud into a soil testing kit to see if the soil is balanced. If not, you may need to add some amendments later.
Remove any large rocks from the soil. Due to the muddy nature of the ground, they should be fairly easy to remove with a shovel.
Rake the lawn so that it is fairly level. If it is very muddy, use the back of a shovel to smooth out rough spots.
Cover the muddy lawn with an inch of sand, and then till it. After it is tilled, cover it with another inch of compost, and then till it again.
Add your soil amendments and some fertilizer. For example, if your soil tested acidic, add some lime. Rake this and the fertilizer into the soil.
Spread the Seed
Disperse the seed onto the lawn. Because the ground is muddy, avoid using a rolling seed spreader. Instead use a hand-held one. Start in one corner or on one end, and work your way to the other end.
Turn a plastic rake upside-down, and use the back of it to lightly smooth and press the seeds into the soil.
Water the lawn for 5 to 10 minutes, 2 to 3 times per day. Do not water so much that you have runoff.