Probably something rudimentary like: evidencing the ability to consistently install frameworks whereby the skill sets of players at their disposal are maximized and things like arriving at solutions during matches are simplified (be it for those who are currently at the club or those who are going to be signed in the upcoming transfer windows). It's why Alex Ferguson got so much out of players who might not have have made it at the highest level of club football under different managers or coaches, and why he could, for example, overcome clubs like Arsenal with hastily assembled midfields...
Trying to make the whole operate at a level that is greater than the sum of its parts, while simultaneously keeping things simple in terms of training and education (so that even the weakest links can play a part), is a rare but absolutely essential quality for a prospective Manchester United head coach — it leads to extended periods of overperformance (every team needs this from time to time), helps you paper over the weaknesses of certain individuals from a collective standpoint (you can't have optimal players at every level of the squad), can come to your rescue during injury or suspension-related crisis (which are inevitabilities), elevates the market value of players that you might want to sell (as opposed to them hitting rock bottom at the club and becoming unsellable), and so forth.
These types of determinations should be made by the Director of Football in conjuction with the Technical Director, the analytical and team management departments, and the Chief Executive (and the ownership if necessary, considering they are the ultimate artbitrators), following a detailed and rigorous assessment period. Managers can be transient figures in contemporary football, someone who is a basically a totalitarian with a “don't give a feck attitude” and walks into a club to initiate the fire-sale of important players can do more harm than good if there aren't appropriate checks and balances in place. They should have no such authority, at least during the early parts of their tenure (when they haven't given a good account of their abilities as a team manager or talent evaluator).