The story I've always heard was that Adm. Grace Hopper coined the term "debugging" when a moth was removed from the computer she was working on. Here's a link to Adm. Hopper's bio, including a picture of the notebook page in which the first bug was immortalized:

http://usnhistory.navylive.dodlive.mil/2014/12/09/grace-hopper-navy-to-the-core-a-pirate-at-heart/
Update: Wikipedia shows the same picture, and says that "bug" and "debug" were in use before the "first actual computer bug." Wikipedia's version of the etymology of "bug" says that the term had been in use in engineering for decades before the taping of the moth, and that the moth was taped into the notebook because those who found it knew the term's meaning in engineering. That seems to make more sense -- taping an actual moth into the log book would have been more gross than funny had the term "bug" not already been familiar to those in the group. In my own experience, Adm. Hopper's name has been associated with "bug" every time I've heard or read this story, probably because she often told the story herself. So, while she apparently did not coin the term, I think she deserves significant credit for cementing "bug" in the programmer's daily lexicon.
Eye-teeth
I had thought the term was hind teeth as well, but eye teeth it is. Wikipedia says:
In mammalian oral anatomy, the canine teeth, also called cuspids, dogteeth, fangs, or (in the case of those of the upper jaw) eye teeth, are relatively long, pointed teeth.
...
In humans, the upper canine teeth (popularly called eye teeth, from their position under the eyes) are larger and longer than the lower, and usually present a distinct basal ridge.
The term eye teeth goes back to at least the 17th century. The earliest I found is in the fascinating 1657 Nature's Cabinet Unlock'd by Thomas Browne:

And the 1660 Lexicon Tetraglotton, an English-French-Italian-Spanish Dictionary by James Howell shows the translations all include eye:

Give [...] eye-teeth
The earliest "give/gave [...] eye-teeth" is illuminating. 1836's The Way-Mark: In Which Some of the Turns of the Broad Road are Pointed Out says:

Rather have [...] eye-teeth drawn [out]
A variant rather have their eye-teeth drawn [out] was said by a Mr. Denny on Wednesday, 17th December 1834 in Congress and recorded in two sources. Once in Congressional Globe says:

And again in Register of Debates in Congress:

And also:

As soon part with [...] eye-teeth
From The Town and Country Magazine, or Universal Repository of Knowledge, Instruction, and Entertainment of 1779:

And 1784's The European Magazine:

We can see the phrase has changed slightly from preferring to give your eye teeth than give/lose money; to the modern giving your eye teeth for something you really want.
Best Answer
The earliest example I found in Google Books is in Hector Urquhart's introduction to 1860's Popular Tales of the West Highlands:
The concept of something having contradictory qualities to different people has been around a long time. "One man's meat is another man's poison" was a 17th century proverb.
Though clearly not the origin of such concepts, here are some 17th-18th century versions. The 1703 The Athenian Oracle: Being an Entire Collection of all the Valuable Questions and Answers in the Old Athenian Mercuries refers to both One's man's pleasure is another's pain and a proverb One's man's meat is another's poison. This book is a bound publication of The Athenian Mercury which ran from 1690 to 1697.
It refers to them in an answer discussing these contrasts and our "perception of what's agreeable to our Natures", or disagreeable. The same could apply to trash/treasure.
Just a few years later in 1706 is Several sermons upon the fifth of St. Matthew:
And the next year, 1707's A General Treatise of Monies and Exchanges three times gives us a similar phrase to the junk:treasure comparison, in the context of the net wealth of the nation/kingdom/commonwealth remaining the same:
Some other variations from the early 18th century:
William De Britaine's 1717 Humane Prudence: "One Man's Fault is another Man's Lesson"
Daniel Defoe's 1719 The life and surprising adventures of Robinson Crusoe of York, mariner: "Thus, what is one man's safety is another man's destruction"
Thomas Brown, James Drake, Aristaenetus' 1720 The Works of Mr. Thomas Brown: "One Man's oversight is always another's Gane."
Nicholas Brady's 1730 Several Sermons: "Estates here often shift their Landlords, and what is one Man's to Day1 is another's To-morrow;"
William Ellis's 1737 The London and country brewer: "But according to the Proverb, One Man's Mistake, is another's Game."
Edward Young's 1741 Love of fame, the universal passion: "How one man's anguish is another's sport,"
A Lady's 1741 The history of Portia: "See what the World is ! one Man's Death is another Man's Joy"
Leonhard Culmann's 1741 Sententiæ pueriles: "One man's weal is another man's bane. One man's wealth is another man's plunder."