If you want to take legal action you would need to refer to the actual legislation and case law that deals with driving licences and tests in particular. The administrative government must comply with the law and you can seek to make them do so through the courts or any administrative appeals tribunal that Ontario may or may not have.
However, before you do this remember: the courts don't give a rat's ass what you "believe". Belief is for churches; courts are only interested in what you can prove and as your relative is making the allegation it falls on him to provide evidence that proves it on the balance of probabilities.
You do not tell us what the submissions were but if they amounted to "I was treated unfairly and wrongly" and "No you weren't" then the appeals boards were quite right to say that he had failed to meet the burden of proof. If independent evidence was needed to change this then it was his obligation to get it and put it before the appeals boards. Remember, he made the allegation of unfair treatment, he has to prove it; DriveTest does not have to prove anything.
You say "... in one email, a DriveTest employee writes that he should not have been driving below the speed limit. Clearly, this is wrong." Clearly it is. However, even if this was raised in the appeals a more simple explanation is that the employee wrote "below" when he meant to write "above". A court will generally prefer an interpretation of innocent error over incompetence; simply because it makes more sense than an examiner genuinely believing that travelling below the speed limit is an issue. Of course, travelling well below the speed limit may amount to dangerous driving depending on the circumstances.
Suing DriveTest as an individual seems financially foolish ...
Yes, principles are expensive; they make lawyers rich!
Thus what can he do?
Suck it up, princess!
... class-action lawsuits, but is this the solution? How do we start or join one?
No. First you need a class of people who have suffered the same harm; while I am sure there are many people who fail their driving tests, most were probably not treated "unfairly and wrongly". Second, you need a case with a wealthy defendant (Ontario's government probably is) and a case which will result in a large settlement so that the law firm who runs the class-action and the financiers who pay for it make their money before what's left is divided among the class.