What to tell professors to persuade them to let a pre-college student work with them
You might want to try aiming a bit lower, such as those schools that don't have graduate programs. 

At these types of places, where it is difficult for professors to even get any students (let alone the top undergraduates) involved in their research program, some professors may actually want to bring on board a high school / junior high student, if for no other reason than to impress some administrative type (dean, department head, etc.) that this particular professor is engaging in some form of outreach (you know, to help spread the word about the program to get students excited about going to school there).

Just send an email to someone involved in the area you are interested in, attach a CV *, and briefly explain your aims. If the school is close by to you, you could also arrange an in-person visit. For the in-person visit route, if you are indeed interested in attending the school for your undergraduate degree, I suggest you indicate this fact when you attempt to setup the meeting. 

Good luck.



* In your CV, you can briefly summarize your self-study progress and plans, your research interests, any projects you've undertaken on your own, and the like, if you don't have any formal educational training in the research area of interest, any papers published, etc.