Elizabeth would go her own way; and never had she pursued it in more decided opposition to Lady Russell than in this selection of Mrs Clay; turning from the society of so deserving a sister, to bestow her affection and confidence on one who ought to have been nothing to her but the object of distant civility. (2.16)
It's odd that Elizabeth, otherwise so snobby, would be positively democratic in her choice of friend. Or perhaps she just likes having someone around who's so obviously inferior in her eyes that she can take advantage of her without complaint.
Anne Elliot, so young; known to so few, to be snatched off by a stranger without alliance or fortune; or rather sunk by him into a state of most wearing, anxious, youth-killing dependence! It must not be, if by any fair interference of friendship, any representations from one who had almost a mother's love, and mother's rights, it would be prevented. (4.3)
Sir Walter's objection on family grounds doesn't sway Anne, but Lady Russell's attitude as a friend does persuade her, since Anne does believe that Lady Russell has her best interests at heart.
"But you, yourself, brought Mrs Harville, her sister, her cousin, and three children, round from Portsmouth to Plymouth. Where was this superfine, extraordinary sort of gallantry of yours then?"
"All merged in my friendship, Sophia. I would assist any brother officer's wife that I could, and I would bring anything of Harville's from the world's end, if he wanted it. But do not imagine that I did not feel it an evil in itself." (8.39-40)
Among the navy men, friendship is so strong that they'll do things for their friends they wouldn't even do for themselves. Compare also Captain Harville having Benwick's miniature, originally intended for his sister, reset for his new bride-to-be – even though it's painful, he would never say no to a friend.
To finish the interest of the story, the friendship between him and the Harvilles seemed, if possible, augmented by the event which closed all their views of alliance, and Captain Benwick was now living with them entirely. (11.12)
Most friends bond over fun activities, but here it's sharing bad experiences that brings the friends closer together. The shared memories of Fanny also seem a factor – together they can talk about a past that few others remember.
There was so much attachment to Captain Wentworth in all this, and such a bewitching charm in a degree of hospitality so uncommon, so unlike the usual style of give-and-take invitations, and dinners of formality and display, that Anne felt her spirits not likely to be benefited by an increasing acquaintance among his brother-officers. "These would have been all my friends," was her thought; and she had to struggle against a great tendency to lowness. (11.16)
Wentworth's close friends make it all the more apparent how much Anne lacks in this area: at this point in the novel all she has is Lady Russell, who's also kind of like her mom, and so it isn't really an equal friendship like Wentworth has with his brother officers.
But the remembrance of the appeal remained a pleasure to her, as a proof of friendship, and of deference for her judgement, a great pleasure; and when it became a sort of parting proof, its value did not lessen. (12.2)
For both Anne and Wentworth, respect is a key aspect of friendship – and so Anne interprets Wentworth's show of respect as a renewing of the friendship between them.
[Sir Walter speaks] "A widow Mrs Smith lodging in Westgate Buildings! A poor widow barely able to live, between thirty and forty; a mere Mrs Smith, an every-day Mrs Smith, of all people and all names in the world, to be the chosen friend of Miss Anne Elliot, and to be preferred by her to her own family connections among the nobility of England and Ireland! Mrs Smith! Such a name!" (17.18)
Sir Walter's making a contrast here between the common and the special: Anne is special, his logic goes, and anyone she hangs out with should be special too, lest she sprinkle her specialness like fairy dust on someone who doesn't deserve it.
She could not endure the idea of treachery or levity, or anything akin to ill usage between him and his friend. She could not endure that such a friendship as theirs should be severed unfairly. (18.19)
It seems like Anne's projecting a bit here – she feels her own relationship with Wentworth was "severed unfairly," so she feels the possibility of something similar happening again especially sharply.
She learned that (the intimacy between them continuing unimpaired by Mr Elliot's marriage) they had been as before always together, and Mr Elliot had led his friend into expenses much beyond his fortune. Mrs Smith did not want to take blame to herself, and was most tender of throwing any on her husband; but Anne could collect that their income had never been equal to their style of living, and that from the first there had been a great deal of general and joint extravagance. From his wife's account of him she could discern Mr Smith to have been a man of warm feelings, easy temper, careless habits, and not strong understanding, much more amiable than his friend, and very unlike him, led by him, and probably despised by him. Mr Elliot, raised by his marriage to great affluence, and disposed to every gratification of pleasure and vanity which could be commanded without involving himself, (for with all his self-indulgence he had become a prudent man), and beginning to be rich, just as his friend ought to have found himself to be poor, seemed to have had no concern at all for that friend's probable finances, but, on the contrary, had been prompting and encouraging expenses which could end only in ruin; and the Smiths accordingly had been ruined. (21.93)
And here's how we know once and for all that Mr. Elliot's not someone you want to keep around: he screwed over his supposed friend Mr. Smith. This puts him in sharp contrast to Captain Wentworth and all the support he's given to his friends Captains Harville and Benwick.
She had but two friends in the world to add to his list, Lady Russell and Mrs Smith. To those, however, he was very well disposed to attach himself. Lady Russell, in spite of all her former transgressions, he could now value from his heart. While he was not obliged to say that he believed her to have been right in originally dividing them, he was ready to say almost everything else in her favour, and as for Mrs Smith, she had claims of various kinds to recommend her quickly and permanently. (24.10)
Anne's friends almost take the place of her family, in terms of the desirable connections she brings to the marriage. By the end of the novel, her chosen connections have a much more important role in her life than the ones she was born with, further showing that birth isn't the be-all and end-all of who a person is.