When little Eppie toddles up to Silas's hearth, you know that something important is about to happen. The hearth (the area in front of a fireplace) is central to Eliot's vision of idyllic country life. Eliot explains how Raveloe is different from Lantern-Yard by describing it as a place "where men supped heavily and slept in the light of the evening hearth" (1.2.1). The hearth represents security, comfort, warmth, and abundance—and not just of food. Hearths ought to be full of children (no wonder deaths by fire were so common before the 20th century), and so when Eppie settles down uninvited on Silas's, good things are about to start happening.
In contrast, Godfrey's hearth—like his sense of responsibility—is cold and empty. He imagines himself "with all his happiness centred on his own hearth, while Nancy would smile on him as he played with the children" (1.15.3), but by letting Eppie be raised on someone else's hearth, he pretty much ensures that he'll never have children of his own.
Only it doesn't seem to be entirely his fault. He's brought up "where the hearth had no smiles" (1.3.38), because his mother is dead and his dad is a jerk. Without that smiling hearth at the center of his childhood, it's a miracle that Godfrey doesn't end up dead at the side of the road, or, as in Dunstan's case, at the bottom of a quarry. Only marrying Nancy—who brings order and peace to his hearth—saves him.
The hearth also seems to be connected to Raveloe's weirdly pagan version of Christianity. Silas is so attached to his hearth that, like a survivalist preparing for the end of the world, he insists on continuing to cook on it even after Godfrey offers to buy him a modern oven and grate. "The gods of the hearth exist for us still" (2.16.30), the narrator says. In Raveloe, the humble villagers might as well be Romans worshiping household gods rather than modern, free-thinking Christians (like Eliot herself, who actually lost her faith after spending time studying the life of Jesus). Home, Eliot suggests, is where the hearth is—but maybe that hearth is something that only exists in the idyllic world of the past. That's kind of a depressing way to think about home.