Here's some Chase family headcanons I've been thinking up:
Disclaimer, it's been a hot minute since I read MCGA, so even though I'm pretty sure none of this goes against canon, if I'm wrong, I apologize.
Annabeth refused to go boating when she first started at camp. Randolph told her when she was around six or seven in excrutiating detail how Caroline, Emma, and Audrey died and she was scared of boats, especially smaller ones, ever since. The first time she ever went canoeing was with Percy a few weeks after he arrived at camp.
The Chase family used to all celebrate Yule together until the rift between Frederick, Randolph, and Natalie. Frederick's family (besides Annabeth) changed to celebrating Christmas on his wife's insistance. Natalie and Magnus still celebrated Yule together, but not as big, though Magnus stopped after her death, and Randolph really didn't celebrate anything after his family's death. (Note: I read a fanfic once where Percy goes to the Chase family's house for Yule, which really made me think on this.)
Emma was an amazing older sister and cousin, and absolutely adored playing with baby Annabeth and Magnus, and kept saying she couldn't wait till they were older. She never saw them grow up.
Frederick, Bobby, and Audrey have/had autism. (Frederick's has basis in canon, which I can go through the diagnosis with canon evidence if anyone asks, but Bobby and Audrey are both on vibes alone). I've also read a couple good fics where Annabeth is diagnosed with autism in addition to ADHD and dyslexia as well.
Annabeth's Stepmother (whom will henceforth be known as SM because she doesn't have a name. Most fanfics call her Helen, though, but I'll stick with SM for now) gave away the family dog when Bobby and Matthew started to walk, even after the dog did obedience school. In addition to the spiders, this was one of Annabeth's deciding factors in running away.
Speaking of the dog, his name was Boeing after the airplane company on Frederick's insistance.
Despite SM's best efforts, Annabeth, Bobby, and Matthew became super close as they got older. Especially when Annabeth starts attending NRU and is super close to them.
Despite being mortals, CJ and New Rome had no problem with the Chase family visiting often though SM never went, and Bobby and Matthew would spend almost every day after school with Percy and Annabeth in one of their dorms or walking around the city. Bobby's dream school is NRU (because of how good their accommations are) and Annabeth is advocating to the school board to let him in, despite not being a roman or greek demigod or legacy himself (just saying, there's a chance they're legacies of Norse gods if Randolph is anything to go off of.)
Frederick did guest lectures at NRU about fighter planes, and celestial bronze/imperial gold bullets.
Magnus and Annabeth used to hold hands when they were drawing or coloring (since Magnus is a leftie and Annabeth is a rightie) during family events. Magnus would always draw trees and Annabeth would sketch buildings she wanted to build.
Magnus keeps in touch with Annabeth and Percy's decendants decades and centuries after they've gone to Elysium. Annabeth had also introduced him to Thalia and Reyna, and the four (plus Alex, of course) would hang out every so often in the centuries to come.
Caroline crocheted baby blankets for each kid when they were born. Magnus' never got finished.
This isn't a hc, but I had the thought of an AU where Bobby and Matthew are Blood of the Pharoahs through their mother's side, and I think that would be awesome because then the Chase family has something to do with the pantheons of all three of Rick's main series.
So, those are it. Feel free to add your Chase family hcs!