"Good evening, good evening," said the Faun. "Excuse me – I don't want to be inquisitive – but should I be right in thinking that you are a Daughter of Eve?" (2.2)
Mr. Tumnus understands Lucy's humanity as part of an extended familial relationship stretching back to her Biblical ancestors.
"My old father, now," said Mr. Tumnus, "that's his picture over the mantelpiece. He would never have done a thing like this." (2.35)
We can tell that Mr. Tumnus is trustworthy and good because he has longstanding family values. It's hard to imagine the Witch keeping a portrait of her father above the mantelpiece!
The White Witch
"Oh, but if I took you there now," said she, "I shouldn't see your brother and your sisters. I very much want to know your charming relations. You are to be the Prince and – later on – the King; that is understood. But you must have courtiers and nobles. I will make your brother a Duke and your sisters Duchesses."
"There's nothing special about them," said Edmund. (4.28-29)
The Witch may only be using them as an excuse, but it's interesting that, even in the tale she uses to trick Edmund into working for her, family represents an important point of appeal. In contrast, Edmund is already willing to cast off his siblings in order to ally himself with her.
She got him to tell her that he had one brother and two sisters, and that one of his sisters had already been in Narnia and had met a Faun there, and that no one except himself and his brother and sisters knew anything about Narnia. She seemed especially interested in the fact that there were four of them, and kept on coming back to it. "You are sure there are just four of you?" she asked. "Two Sons of Adam and two Daughters of Eve, neither more nor less?" (4.20)
Both the local construction of the Pevensie family – four children, two boys and two girls – and their historical genealogy – descendants of Adam and Eve – makes them the perfect group to fulfill ancient Narnian prophecies about the destruction of the Witch.
"It is a lovely place, my house," said the Queen. "I am sure you would like it. There are whole rooms full of Turkish Delight, and what's more, I have no children of my own. I want a nice boy whom I could bring up as a Prince and who would be King of Narnia when I am gone." (4.26)
The Witch's barrenness – her inability to generate her own offspring and become a mother – is part of her evil. She is not a functioning element in a family group, but a dangerous outlier.
Mr. and Mrs. Beaver
"The potatoes are on boiling and the kettle's singing and I daresay, Mr. Beaver, you'll get us some fish."
"That I will," said Mr. Beaver. (7.46-47)
Mr. and Mrs. Beaver are the ideal Narnian couple. While Mrs. Beaver attends to the sewing and cooking, Mr. Beaver provides the food and keeps the house repaired. They're very much out of the time when C.S. Lewis was writing – the late 1940s, post-World War II.
"I tell you he is the King of the wood and the son of the great Emperor-Beyond-the-Sea." (8.22)
Just as Jesus is the Son of God, Aslan is the son of the mysterious and powerful Emperor-Beyond-the-Sea. It's important that we see Aslan as taking on a familial role; unlike the Witch, he is not a stand-alone creature, but part of a family system.
"Don't go on talking like that."
"Like what?" said Susan; "and anyway, it's time you were in bed."
"Trying to talk like Mother," said Edmund. "And who are you to say when I'm to go to bed? Go to bed yourself." (1.5-7)
When the four Pevensie children are sent to the countryside to get away from the air raids in London, older sister Susan tries to reposition herself as a surrogate mother.