Paul's mother defines luck this way, when Paul asks her:
"It's what causes you to have money. If you're lucky you have money. That's why it's better to be born lucky than rich. If you're rich, you may lose your money. But if you're lucky, you will always get more money."
She claims her husband is not lucky, though she obviously has more than her share of riches--clothes, house, servants, stables...and more. When she does get more money, it's never enough. Apparently she is the one who is not lucky.
The post above is correct about the eyes--the descriptors throughout the story depict a more and more obsessive/crazed/possessed/determined/out-of-control boy. This is one of the most striking images of the story and a reflection of Paul's intensity leading to his death.
Luck is confused with money in the story. The mother says to Paul that they are poor because they do not have enough luck. Paul then sets off on his self-destructive course to gain more luck or money.
Eyes are repeatedly referred to in the story to describe and chart Paul's progression into sickness and finally death as he becomes "possessed" by the spirit of the rocking horse that gives him the names of winners. Re-read the story and trace this imagery through and note how a supernatural tone is created.
1. Luck is chance, in my eyes, and one has to be in the right place at the right time, etc., for it to happen. Luck, to me, happens when you may not deserve it or expect it, whether it is bad luck or good luck.
2. The eyes are the windows to the soul, as the saying goes. They can often reveal if people are telling the truth or lying, what kind of mood they are in at a given time, and if they are sick/ill.