The folksong "Drill Ye Tarriers Drill" is well-known:
Wikipedia Drill Ye Tarriers Drill.
The title refers to Irish workers, drilling holes in rock to blast out railroad tunnels. It may mean either to tarry as in delay, or to terrier dogs which dig their quarry out of the ground,1 or from the French word for an augur, tarière.
But the word appears to be used in a different sense in Stephen Crane's "The Pace of Youth" (Crane, Stephen and Gado, Frank. Drawn from Life. Signature, 1997).
"That young tarrier," he whispered to himself. "We wants to quit makin' eyes at Lizzie. This is too much of a good thing. First thing you know, he'll get fired."
In this context, the tarrier is a young man who works in an amusement park assisting with the merry-go-round. Clearly he's not drilling holes or laying railroad track.
Yet the Oxford English Dictionary on-line
OED through VPN of Virginia Commonwealth University library
recognizes no such sense of the word. Noun 1 is one who tarries; noun 2 is "A boring instrument, an auger; now, an instrument for extracting a bung from a barrel." Perhaps this is related to the railroad workers, who are drilling and driving stakes, I think.
Merriam-Webster is even worse, bouncing immediately to the verb and also suggesting "tarry" means "covered with tar."
I am mystified that a well-known word—at least in American English—would not show up in either dictionary, when "navvy," for instance, shows up in the OED.
If "one who drills" is the only meaning besides "one who tarries," perhaps the use in the Crane story is obscene—meaning, "that boy who wants to do it to my daughter." But that seems a stretch?
Best Answer
In the Crane story, the speaker may be comparing the young man to a (tenacious) terrier. It seems to have been a not-infrequent alternate spelling judging from Google Books.
How many times have we heard or read "If I catch you sniffing around my daughter"?