I will be starting high school next year, and I have the option of going to two different schools. The first is a prestigious public high school that is completely focused on academics, and will likely be very difficult to excel in. The other option is my local high school, which will offer a greater variety of courses and extracurriculars, but is comparatively easier in terms of grades and competition.
Will I have better chances of getting into an Ivy League if I attend, and do well at a prestigious school, or if I’m a top student at my local school?
Your chances of admission at top tier universities won't differ significantly based on which high school you attend. This choice is more about which school will give you the best opportunities to pursue your passions and build a great application.
For example, are there classes at the local high school that you really, really want to take that aren't offered at the academically-focused public high school? Are there extracurricular opportunities at the local high school that you plan on seriously pursuing that are not available at the other school? If so, the local high school might lead to higher chances of admission because you'll be able to showcase your talents better given the course and extracurricular offerings.
If, on the other hand, you don't see any classes or extracurriculars that you really care about that are only offered at the local school, the academically-focused public school could be a better option. You might also choose that school if you think being around other students who are more intense and focused will help motivate you. Make sure you take your personality into account!
Essentially, neither school will give you inherently better chances. Take a serious look at both options, and consider which has greater potential for helping you explore your areas of interest and access opportunities that are important to you.