"act of worrying a chained or confined animal with dogs," c. 1300, also figurative, verbal noun from bait (v.1). Related: Baitingly.

c. 1300, wirien, "to slay, kill or injure by biting and shaking the throat" (as a dog or wolf does), from Old English wyrgan "to strangle," from Proto-Germanic *wurgjan (source also of Middle Dutch worghen, Dutch worgen, Old High German wurgen, German würgen "to strangle," Old Norse virgill "rope"), from *wergh-, from PIE root *wer- (2) "to turn, bend."

The "strangle" sense generally was obsolete in English after c. 1600; the figurative meaning "to annoy, bother, vex" is by c. 1400. Meaning "to cause mental distress or trouble" is attested from 1822; intransitive sense of "to feel anxiety or mental trouble" is attested by 1860.  Related: Worried; worrier; worrying. 

1901, "stupid or foolish person," probably a shortening of muttonhead (1803) in the same sense; see mutton and compare meathead, etc. Mutt was used by 1898 of a dog, especially a stupid one, and perhaps this is the same word formed independently (muttonhead also was used of stupid animals), or else a separate word of unknown derivation. Wright's "English Dialect Dictionary" (1900) has "Mutton! used in scolding a dog, prob. in allusion to the offence of sheep-worrying."

"That dog ain't no mutt," McManus would say as he stood behind the bar opening oysters; "no an he ain't no rube! Say! he's in it all the time when Charley trims the steaks." [Robert W. Chambers, "The Haunts of Men," 1898]

Used by 1910 in dog fancier publications to refer to a non-purebred animal.

Mutt and Jeff is by 1917 in reference to "a pair of stupid men, affable losers," or to one tall (Mutt) and one short (Jeff), from the comic strip characters from the heyday of the newspaper funny pages, Augustus Mutt and Jim Jeffries, in U.S. cartoonist Henry Conway ("Bud") Fisher's strip, which debuted in 1907.