Why is Brienne so loyal to Catelyn? Sure, she was nice to her (unlike 90% of the time) but it was basic courtesy, I don’t see why she’s risking her life out there for Sansa (even though she’s 100% more deserving of Brienne’a service than Catelyn ever was)

turtle-paced:

Because Catelyn’s treatment of Brienne went well beyond basic courtesy.

When Brienne was being outright attacked, suspected of murdering Renly, Catelyn stepped in on her behalf.

Her own voice sounded wild and crazed to her, but the words poured out in a rush as the blades continued to clash behind her. “A shadow with a sword, I swear it, I saw. Are you blind, the girl loved him! Help her!” She glanced back, saw the second guardsman fall, his blade dropping from limp fingers. Outside there was shouting. More angry men would be bursting in on them any instant, she knew. “She is innocent, Robar. You have my word, on my husband’s grave and my honor as a Stark!”

That resolved him. “I will hold them,” Ser Robar said. “Get her away.“ 

– Catelyn IV, ACoK

And then Catelyn actually does get Brienne away from the camp. Catelyn saved Brienne’s life. That’s a pretty good start in terms of Brienne feeling loyalty to her, yes?

Furthermore, Catelyn doesn’t treat Brienne like a joke. She takes Brienne seriously when Brienne says she wants revenge on Stannis – her objections to Brienne’s revenge plan are that it’s futile and she’ll only get herself killed, not that she should leave it to a man. She accepts Brienne’s oath of service to her as she would have accepted a man’s oath. Later, Catelyn makes active use of Brienne’s skills when Catelyn entrusts her with Jaime’s wellbeing on the trip to King’s Landing.

Catelyn wasn’t just saying please and thank you and refraining from laughing at Brienne. She saved Brienne’s life. She was told Brienne that her feelings were valid, she treated Brienne with respect as a warrior, and she backed up her words with actions in giving her important, worthy work to do. Given that people not laughing at Brienne is a social encounter of note for her, I think it’s easily understandable that Brienne’s so fiercely loyal to a woman who treated her as if her life meant something and her skills were valuable.