"Give someone the eye" does mean, as you state, to look at somebody with a more or less open display of romantic interest.
"Give somebody the big eye" has 428 examples in Google. 424 of them show the lyrics of "Blues in the Night," of which these words are a part:
My mamma done tol' me,
When I was in knee pants,
My mamma done tol' me, "Son!
A woman'll sweet talk, and give ya the big eye,
But when the sweet talkin's done,
A woman's a two-face,
A worrisome thing who'll leave ya t' sing
The blues in the night"*
In this context, it doesn't seem that "give ya the big eye" means exactly to display a romantic interest, although that is implied. It seems to mean that a woman will look at a man with big eyes – that is, very wide, open eyes – in order to look innocent and then deceive him.
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Other examples of "big eye" include this. The person could "give someone the Big Eye" for the purposes of observation. But this expression is original with the interviewee:
He pounces on each kid who doesn't see him coming, sweeping them out of the hall with a rapid fire of hectoring, friendly teasing and, if necessary, what he calls "the Big Eye." He learned the Big Eye from his mother and when he uses it on a kid, it can mean anything from "I see what you're up to" all the way to "You cut that out right now."http://www.siliconvalley.com/mld/twincities/news/columnists/5289639.htm
Another includes this, which means....to be grateful to have escaped from harm???:
My prayers are with all of the people...hoping none are suffering injuries or loss of their homes tonight ! We slid right between 2
tornadoes , and that's enough to give me the Big Eye, but I think I must try to catch some zzzzzzz's.....Jeannie T
http://archiver.rootsweb.com/th/read/APPALACHIAN-LIFE/2003-05/1052119196_______
"Give somebody the eye" is a fairly frequently used expression, but "give somebody the big eye" seems almost non-existent. Yuri, where have you heard / seen this expression?
It may be that "give somebody the big eye" is in lexicons other than my own American. Please let us know!
Rachel
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*"Blues in the Night," a song by Johnny Mercer and Harold Arlen.