I'm not sure if this has been mentioned yet but by the time we were picking a MS with my DC, she was really one driving the bus. She and I sat down and we put together a system for comparing schools. Out of about 8 factors (that I can remember) only one was something I insisted on being on the list and that was location. I was also the one concerned with tracking (putting advanced or general kids and etc. in different classes). Other than that, DC was concerned with things like where her friends were going, school size, academic reputation, homework expectations, interesting opportunities, whether the school went through to HS (I cared about this as well), school culture, lockers (yes, this was a factor, ha, ha!), uniforms and just her overall impression when vising.
Behind the scenes I also looked into things like administration and teacher retention, test scores (and whether schools had a reputation of teaching to the tests), and all the rumor mill type stuff that I picked up in the process.
FTR, we were only looking at public schools (and there is a great deal of choice for students in our district). I imagine if there was the option for choosing private, we would have a broader set of criteria.
Now that DC is mid-way through 7th I think important factors for her personality are: high expectations, unique opportunities, good classroom management, prioritized love of learning, diversity in students and staff, kind teachers, a size that is not too small or too large, a school that supports kids to get into a highschool of their choice (in my area that means supporting kids in testing), support for learning outside of class (coach class and etc.), a school that has a nice balance of support and growth in this interesting time of development.