(i) We learn in Chapter 11 of Little Dorrit that Break of Day is an inn of some sort.
(ii) The phrase "Break of day" = dawn.
(iii) A godfather is defined by the OED as
1.a. A male sponsor considered in relation to his godchild; a male godparent.According to the practice of the Roman Catholic, Orthodox, Anglican, and some other churches, certain persons (commonly two at least, a man and woman) assist at the administration of baptism, make profession of the Christian faith on behalf of the person baptized, and guarantee his or her religious education.
In extended meaning, and indicative of a man who gives guidance and direction to something, we have
2.c. Chiefly with the and of. A man who founds or is otherwise influential in a (specified) organization, movement, or field.
1973 Melody Maker 24 Feb. 57/4 Another name's been added, just to keep things modern. ‘James Brown—The Godfather Of Soul.’
2012 Jrnl. Tribune (Biddeford, Maine) 29 Mar. a5/2 He [sc. Larry Stevenson] basically was the godfather of skate culture. Before him, skateboards were toys.
[It is worth noting that, despite the OED, in this meaning, there is a nuance: “The godfather of X” tends to indicate not someone who originates a organization, movement, or field - that would be “the father of” - but more someone whose influence is significant in raising popular interest. However, this distinction is rarely made.]
Thus we have:
"The Godfather Break of Day was peeping at its namesake." = "Dawn was looking at the inn."
Dawn is the original object after which the inn was named. Dawn is the thing that gave the inn its identity.