The biggest thing that would be helpful and also easy to collect (ignoring politics or regulatory issues related to tying identifying information to test scores) would be scores grouped by socioeconomic status and parental status (e..g, married, divorced, never married, employed, unemployed, highest level of education, etc). But I can think of lots of data that would be nice to have if I ran the world. IQ scores and proficiency scores from before or right after kindergarten, 4th, 8th, and 12th grades would be interesting. Hell, I'd throw in BMI, height, and race and anything else that can easily be measured, even if it might only be of interest to researchers and not parents. And parental IQ if I was king for a day.
But being realistic, I think what decile the household income of the child's primary residence falls in would get you 85% of the way there without having to do anything other than have all households fill out the same forms required to qualify for free or reduced lunch.
Which again, may not be politically easy, particularly if parents don't want to provide that info (which I certainly wouldn't want anybody at the school knowing my financial info), but would be really easy data to compile ignoring that.