A rough estimate of the answer to a lengthy or difficult computation. Usually worked out in your head, on a cocktail napkin or on the back of an envelope.
A general principle or likely tendency.
Etymology: Unknown...please fill it in if you know where this expression comes from.
"They" say that the origin is English common law from several centuries ago. A man brought before the court on a charge of beating his wife would be acquitted if the rod was thinner than his thumb - the so-called "rule of thumb".
I have no idea whether or not this is true.
Urban legend - from the tip of my thumb to the first joint is almost exactly one inch - I measure with it now and then, especially on printed maps.
Yes, myth: http://www.debunker.com/texts/ruleofthumb.html
Couldn't it be that habit of the painters - when drawing outdoors some landscape, to stretch their hand with a thumb up to "measure" a distant object, and then transfer it to the canvas? I'm not sure, it's just a thought.
Examples: LittlesLaw, RuleOfSeventyTwo, PiSeconds
See HeuristicRule.