"Giving someone the hairy eyeball" is an American English slang expression for staring at someone in an openly suspicious, hostile, confrontational or demeaning manner.
To look through narrowed eyes.
To inspect something thoroughly, with scepticism. e.g. "I'll give that report the hairy eyeball this weekend"
Related concepts are:
To look askance at something, except that the hairy eyeball is full-on, not sideways.
The evil eye or mal'occhio is similar but is superstitious.
See also: scrutinize, scrutiny
The hairy eyeball is not the most common of expressions; many people have not heard it. The hairy eyeball is claimed to be common slang in the 1970s, but it is also mentioned in the song Alice's Restaurant (1969).
There is now a "Hairy Eyeball Ale" made by an American microbrewery called Lagunitas, in Modesto, California (http://www.lagunitas.com/beers/index.html). It is 8.8 Alcohol. Now that'll make your eyes feel hairy the next morning.
If you took the other writeup as fact, you need to give it the hairy eyeball. In fact, go to your local bar and ask for a "hairy eyeball". All you'll get is the hairy eyeball.