What would be the desired qualities of an interested Mathematics major a Top tier university want? I understand that this question is a bit too vague and vast, but generally what’s the trend?

I understood that good grades in math, aps ( I can’t give APs, so I think I will do Stanford ULO if I get in) are required. But what qualities would make them feel/show like “Aaha this is passionate for maths?”

I think camps, olympiads would probably make? What would be your thoughts about research/mentorship? What about Stanford ULO? How about having a Math blog? Or making books/ handouts or teaching maths. Any comments?

In general, you should take the most rigorous courses available in your school, that you can perform well in. From the Stanford ULO website, it looks like Calculus AB and BC are pre-requisites for their advanced math courses, and Physics C EM is a prerequisite for the Physics coursework. So long as your high school coursework adequately prepares you for those courses, taking them could likely improve your application.

All of the options you listed (research/mentorship, a math blog, etc.) could help demonstrate your passion for math and your interest in majoring in math.

Given how competitive top-tier university admissions is, and how many variables affect how they would view each applicant, it is impossible to say which extracurriculars would be better for which schools or otherwise handicap a particular application.