You'll always want to keep it covered, anyway. With a tarp over it, it is just a curiosity; you can't actually tell what it is.
If you want to keep your mud pit viable for a while, you'll need to line it before you fill it, and cover it like you would a bowl of pudding (to keep the surface from drying and to keep leaves and other debris from collecting in it).
If you don't line it, it will dry out from the bottom up. Also, you need to be able to stir the bit thoroughly, from time-to-time, so it's better not to make the pit really deep. About 3 feet deep is practical for maintenance. Going deeper is really just a waste of good mud; it takes a lot of mud to fill a pit that is 6 ft (2m) deep and 4-6 ft (1.3-2m) in diameter ... a LOT of mud.
Looking4messyFun said: Any particular mix better than others.
What is your natural soil like? Is it rocky/gravely? Do you have high clay content? Sandy? Loamy?
Loamy soil or soil with a lot of clay are both good. If your soil is just sand or gravel, you may have to import something. You can mix loamy stuff with sandy soil for decent results. Gravel is pretty much useless, so you'll just have to dig it out and import a lot of potting soil, clay, etc. There are lots of different kinds of clay, but, if you're buying it, you may only have certain types available to you, locally. Some nice loamy potting soil makes pretty good mud. You should buy some different kinds and find out which one has the finest mix. Some potting soil has a lot of course material.
Also, if you have very well drained soil, you definitely need to line the pit with something. All of your finest, most desirable content will leach out with the water. All you will have left is the coarse material you didn't want in the first place.
p.s. be wary of people who tell you to bury a garden hose at the bottom of your pit. That doesn't work, but the idea has become an urban legend.
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