All I'm seeing right now is James trying to appease Marilla after he becomes sort of tenuously between friends and something more with Lily, and the common room is empty, and he's kneeling in front of a couch, and waggling a little fake mouse in front of a very unimpressed tabby. He gives up on trying to bribe Marilla with toys and instead expresses his interest and intentions towards Lily, and at the end Marilla licks his forehead, which he is delighted about (not disgusted because Sirius, duh)
And then he and Marilla become quite close, and the first time, Lily freaks out because she can't find Marilla and goes to enlist the Marauders help- because they have this eery way of knowing where everyone is- only to find Marilla curled up in James lap, purring rather loudly while the boys play exploding snap. Fast forward a few years, and Lily and James are happily married. There are many jokes of 'you love my cat more than me' 'well she's certainly more playful' and when Harry is born, she spends most of her time curled up in his nursery, protecting him from all the things that go bump in the night- or, at least, mice. She particularly likes James and Harry's hair, she likes burrowing her face in it, she likes biting it- which James scolded her for once so she doesn't do it to Harry- and she also likes playfully knocking James' glasses off his face, either by rubbing against him or by batting at them with a paw. When Voldemort comes a'knocking, Marilla has been locked out for knocking Harry's baby food onto the floor quite maliciously. For a few minutes beforehand, she could be heard hissing and yowling (cos cats always know whats up), and when Voldy turned up, she left some nice scratches in his cloak. After the door was left wide open, Marilla entered the house, waited by James side for a few moments, batting at his glasses in an attempt to get him to react, then went upstairs when she realised he wasn't going to. She found Lily in the nursery, curled up by her side and hissed at anyone who entered the house and came between her and Harry. When Snape entered, Marilla hissed quite exuberantly, though moved when he pulled Lily into his arms, and moved to protect Harry more definitively. After Harry was taken away, Marilla found herself without a home, and spent her days roaming around Godric's Hollow, feeding off mice and anything Bathilda Bagshot would leave her, and lived out her days in the wild- she spent her last night curled up in front of Lily and James' tombstone, and it is uncertain whether old age or the cold was the cause of her death.