Green’s Dictionary of Slang

1. selfish, domineering.

1929

M. Beckwith Jamaica Proverbs (1970) 30: Strong-eye pickney neber hab right.

2. determined; thus make/play strong-eye v., to act determinedly.

1916		J.G. Cruickshank Black Talk 31: To play or make, ‘strong eye’ is to play a bold game.
1873		C. Rampini Letters from Jamaica 87: A determined person is ‘Mr. Strong-eye’.