I had no luck finding either definition of picture of the mess kid, which was evidently a small pail with vertical sides and probably a cover. The fore-runner of the meskit I had in the Army. Here's a URL that shows pictures of galleys, and mentions kids without describing them.
http://ic.net/~blues/galleys.htm

I finally found a place where he called the "kids"
"wooden tubs from which the sailors eat". They had to be very nicely made to keep clean, and I wonder how they could spare water for it, unless they were washed with salt water.

I found a place near end of book, in which Dana said that the "kid" was a tub holding all the food for one watch. There were no plates, and no cutlery except their own jacknives or sheath knives. So everybody put his dirt;y hands into the tub. YUM,YUM. They did have individual quart
tins for their tea.