Gah, this post has forced me to delurk.
Those three schools are ones that I am intimately acquainted with. All three of them are used to young college students, and a 15 year old will not be terribly out of place. On the young side, but possibly not even the youngest.
As a mother, I would not send my 15 year old to Caltech. There just isn't enough adult supervision and very bright kids can get up to unhealthy sorts of insanity. A lot depends on what Hovse a student sorts into, and some are more sane than others, but I wouldn't want my child there at 15.
Harvard and MIT both do well with supervision for Freshmen.
Of the two schools, I would allow him to decide. Harvard is easy. It can be academically challenging, but he should be able to get straight As with very little effort and he will have a lot of free time to spend on extra-curriculars. A trap many undergrads at Harvard fall into is killing themselves with the extracurriculars and doing none of them well. The one exception to easy As is the engineering department where there isn't any grade inflation. The Freshmen dorms have folks looking out for the kids.
MIT will be challenging, and all his classmates will be challenged, even the ones who are used to breezing through. At some point every student at MIT feels stupid (same with Caltech, but there's less adult support). A common trap is some kids with previous preparation do just fine their first year but never learn to study and crash and burn spectacularly their sophomore or junior year (also a problem at CalTech), eventually being outstripped by kids who were struggling their first year. MIT is a much more stressful environment than Harvard. BUT they have probably the most comprehensive residential support system of any school I have seen, possibly any school anywhere. They have to. They put enormous effort into keeping their suicide rate the same as the national average for that age group and there will be someone looking out for your kid. They even have "suicide days" once per month when they insert a no classes holiday because months without holidays have more suicides on average. There are many points of referral for mental health counseling through the residence, classes, and so on. People will notice if your kid disappears. Adults will also keep them from blowing themselves up, for the most part (not true at Caltech). Kids at MIT generally have one extra curricular activity in addition to their dorm activities. If he does go to MIT do not allow him to sort into Senior House (the drug dorm). There are drug floors in the other dorms as well that you will want to put your foot down about if he chooses MIT. East campus dorms tend to be more quirky and West campus dorms tend to be more "normal." The frats are stereotypical frats and I wouldn't let my kid in one at 16 or 17.
If it were my son going I would lean towards Harvard, but make sure he knows what he's getting into at MIT if he wants to go there. Also, although there's a lot more quirky types at MIT than Harvard, both campuses have both socially normal folks and quirky folks. It's easier to have friends while socially awkward at MIT, but not impossible at Harvard, and vice versa for socially ept. There's probably more room for mental growth at MIT, but there's plenty of time for growth in graduate school... Harvard is a good transition. If he's just thinking masters, MIT has a great 5 year MEng program. And if MIT academics get too tough, he can always drop down to course 15 (business). Though Harvard also has better financial aid, as I'm sure you've noticed.
