Silver is soft-- but only for a handful of people. He's a fearsome pirate with his own reputation, while he's had some semblance of honor, he's not been above letting folks die or get hurt for his own ambitions.
That boy changed him-- brought out the best in him. A side of himself that Silver was unsure he'd ever see again. Some secret, lost part of him he buried after the loss of his own family years ago. Jim needed guidance, and Silver had been tasked with teaching him.
For the sake of his goals and ambitions he took that boy under his wing, playing the part of the ship's cook taking charge of the cabin boy. Then he got to know the kid-- see him for who he was. Stubborn to a fault but with a heart of gold.
Silver saw so much of himself in Jim-- so much of the future he'd hoped to see with his own pup-- it changed him for the better.
Despite himself-- and all the decades of work he'd put toward his dream-- he threw it away in an instant for that ratty little cabin boy of his.