The rising eye is the Lyte family's crest:
This was an eye rising up from the sea. From it rays of light (or lashes) streamed out, half-covering the surface of the cup. It was this emblem Merchant Lyte had on everything he owned—carved above his counting house on Long Wharf, engraved on all his silver—even on dog collars and harnesses. Miss Lavinia had it stamped on her Spanish-leather gloves. Johnny knew it was cut on the slate gravestones of the Lyte family on Copp's Hill. (1.6.4)
The rising eye is a symbol of the power and influence of the Lyte family, and at the beginning of the novel, Johnny is fascinated with the symbol just as he is fascinated with the family. However, by the time the rising eye makes another significant appearance, his feelings have changed:
They went up the steep road from Milton. It was here Mr. Lyte had his country seat. Then Johnny got out, struck tinder, and lighted the lantern he had found in the chaise. He stood by the entrance gates. Yes, Cilla was right. They had smashed the arms carved upon the gates. The poor people of Milton had had enough of that rising eye. Johnny wasn't sure but he had as well. (8.2.3)
At the beginning of the novel, the rising eye inspires pride and longing in Johnny, but after the British occupation begins, he matures to the point where he no longer needs to take pride in that image. He realizes that the Lytes have nothing to give him and that he needs to make a life for himself without their coat of arms because he no longer draws any part of his identity from that relationship.