shrubs and small trees of the genus Cornus, 1610s, earlier dog-tree (1540s); the first element sometimes said to have been perhaps dag -- compare dagger and dag (v.) "to pierce or stab" (1630s, perhaps 15c.).
The trees have hard, white wood that was said to have been used in making butchers' skewers; another name for it was skewer-wood. This explanation is as old as the word itself in English, but the form *dagwood is not attested. Another guess is that the tree was given the name in reference to its fruit, which was called dogberry from 1550s, and dog (n.) had implications of "cheap, inferior" (i.e. "fit for a dog").