A dewclaw is a vestigial digit on the foot of some mammals, notably dogs.
Here's a picture of a dog's paw with an arrow pointing to the dewclaw.
The claw part is obvious enough, but why dew?
I'm afraid that no one knows for sure. The word is some four hundred years old. Dictionary.com maintains that it may be related to the kind of dew that appears overnight, but it cannot fortify the claim.
Dewlap is further down in the same entry:
Dictionary.com is virtually identical on dewclaw (it refers to dewlap too), but there is a little more in its origin entry on dewlap:
YourDictionary.com cites American Heritage Dictionary, 4th Ed. for a claim that "dew" may be an alteration of "toe", but it also says prob. so called because it does not touch the ground, but only the dew on the surface.
Dog or dew may be cognate with Scots dewg/Old Scots duig, always used in the plural and having the meaning "small pieces, shreds, fragments" (see Dictionary of the Scots Language), suggesting "fragmentary claw". This connection was proposed in a thread on the subject at The Straight Dope.
The OED says