Green is a colour associated with sickness, possibly because people's skin sometimes takes on a slightly yellow/green tinge when they are seriously ill. Green is also the colour of many unripe foods that cause stomach pains.
The phrase was used by, and possibly coined by, Shakespeare to denote jealousy, in The Merchant of Venice, 1596:
How all the other passions fleet to air,
As doubtful thoughts, and rash-embraced despair,
And shuddering fear, and green-eyed jealousy! O love,
Be moderate; allay thy ecstasy,
In measure rein thy joy; scant this excess.
I feel too much thy blessing: make it less,
For fear I surfeit.
In Othello, 1604, Shakespeare also alludes to cats as green-eyed monsters in the way that they play with mice before killing them.
O, beware, my lord, of jealousy;
It is the green-eyed monster which doth mock
The meat it feeds on; that cuckold lives in bliss
Who, certain of his fate, loves not his wronger;
But, O, what damned minutes tells he o'er
Who dotes, yet doubts, suspects, yet strongly loves!
See other phrases and sayings from Shakespeare.