Although you don't provide your son's high school stat's (gpa, ranking), I am going to assume he is at least a top 10%, if not top 5% student based on the info provided. 1500 is a great score, but for Yale, it is going to be in the 25th to 50th percentile, so below average. You did not post his SAT2 scores, which while not required, are "recommended". The legacy advantage is minimal, if any. Remember he will be compared to some extent against all the other legacies, who likely will have all of the high SES advantages of going to great schools, resources for test and other academic prep, opportunities for all types of EC's, etc.... The only "tangible" advantage for a legacy is a guaranteed second reading of his file. If you and your spouse are consistent donors and volunteers, you might be able to get Development to add a note in his file. Remember around 80% of legacies are rejected. That's better than the general pool as a whole, but the quality of the legacy pool is also higher.
So how do you improve your son's chances?
1. Make sure he continues to do well academically this year and does not develop senioritis and drop off his senior year in grades and rigor of courses.
2. Others may disagree and say 1500 is enough to get you in the door of serious applicants, but since Yale superscores the SAT, I'd have him take it at least one more time to try to get into an above average band.
3. But most importantly, he needs to develop a coherent story about himself through his essays, LoR's and portrayals of EC's. It sounds like he is moving around in schools, so make sure he develops a good, personal relationship with at least 2 teachers this year who will be able to write strong LoR's. The goal is to have these pieces of the essays, LoR's and EC's present a picture, quoting from the Yale Admissions page, of an applicant “Who is likely to make the most of Yale’s resources?” and “Who will contribute most significantly to the Yale community?”
You might hear advice that candidates have to be "spikey" -- no, if your kid is spikey, that is great and highlight that. If he is round, celebrate that. What you don't want to do is to manufacture a picture of candidate of what you think Admissions is looking for that is dissonant with your son's actual interests and accomplishments. An application that raises questions because of internal inconsistencies will hit the reject pile quickly (paraphrasing a comment that the Dean of Admissions made recently).
Good luck. Your son sounds like a very accomplished kid.
Depending on the institution, you will be asked to fill out certain forms (e.g. FASFA and CSS) and provide certain documentation (e.g tax returns). Link to MIT page https://sfs.mit.edu/undergraduate-financial-aid/aid-info#320 For schools that offer merit scholarships, there are likely separate forms to complete and submit. For schools that accept your son, they will provide a detailed financial aid package at that time (how much, what type). Unless you applied ED (in which case you are stuck with that school -- MIT is EA), you definitely should compare packages and you might even have the opportunity to get better aid by showing a superior aid package from one school to another and asking the second school to "match", especially if they are "peer" schools competing for similar students.
I normally believe that doing a lot of practice tests (whole and by section) along with using resources like Khan Academy and prep books (start with the College Board one) is sufficient. But here, since you have been out of school and you seem to be lost on certain types of math problems, you should consider a tutor if you can afford one (perhaps a local high school math teacher who is willing to help you at a reasonable hourly rate). A good tutor should be able to spot the types of problems and concepts that you are having difficulty with and address those with you specifically. If you want to increase your math score by 100-200 points, you have to be familiar with all concepts that will be tested. As @intparent points out, there are repeating types of problems (format and/or concept) and one of the keys to high scores is immediately identifying the "type" and then solving for the answer through the process you have developed for it. If you are consistently baffled by certain types of questions or concepts, you will not be able to raise your score to your targeted level.
Was on an interesting conference call the other day with Yale's Dean of Admissions, Jeremiah Quinlan. He provided some interesting insights on the process that I think posters here often speculate about. The process he describes is similar to the Wesleyan process described in the "Gatekeepers" and even in the film "Admission".
The Yale admissions officers are each given a region. The regions are at least largely geographic because he spoke in geographic terms and probably defined by the number of applications they anticipate receiving from that region to even out the workload. After a first sorting by the officer responsible for the region, there is a second reading to get the list down to candidates that would go before the Committee. There was no suggestion that each region got an equal number of candidates in to the next round. Of the 33,000 applicants for the class of 2021, he described 20,000 as academically qualified. By the time they got to the Committee, the list was culled to about 6,000. So yes, while candidates are considered in pools of various sorts, the first 14,000 cuts of qualified candidates is done primarily by 1 or 2 individuals that are looking through a geographic pool. I am sure they have subpools in mind when they go through their portfolio, but when you deal with decisions made by 1 or 2 persons, human subjectivity will be a bigger factor since the decision is not based on a large group consensus. As they go into Committee, each Regional officer is tasked with prioritizing their list of candidates who are being considered and is the advocate for those candidates. It is not hard to imagine that there are candidates which fly through the Committee by dint of their accomplishments and how they presented themselves through their essays, LoR's and EC's (these are likely the kids who get admitted into multiple highly selectives). I suspect it is at the next level of candidates when the Committee is trying to build their "ideal class", that comparison based on subpools becomes a larger factor. I would also hazard a guess that based on the sheer number of applications each officer has to go through in the first round, the first sorting (conscious or unconscious) will be based on objective stat's, and the further you fall down that curve, the greater the need for something in the subjectives to jump out to move on.
There is very little, if any, admissions advantage in applying EA. Having a good upward trend in rigorous courses senior year will be a plus. EA however is a great tool in potentially cutting down applications your son would need to complete (including associated fees), reducing some level of stress, and gauging the strength of his overall application, including perhaps getting a read on the strength of his essays and LoR's. My general advice is to apply to your state flagship (or other high quality institution with rolling admissions) early, and apply EA to schools that are high matches/low reaches in your son's situation since he can improve the strength of his application between now and the RD deadlines. Some EA's are restricted (can only apply to 1 school EA), so you will need to gauge whether you want to cast a broader net or whether that particular school is high enough on your son's list that it could potentially remove most other schools from his application list.