There is one main reason why Louis Riel was executed (and why it was right for him to be executed). The reason is the fact that he committed treason. It is very hard to argue that Riel did not commit treason.
Treason is the crime of betraying your own country or of trying to overthrow its government. There is no question that Riel did the second of these things. The rebellions that he led were aimed at creating an independent nation that would be made up of lands that were part of Canada. Going to war against your own government for this purpose is clearly treason and treason is typically punishable by death.
The only real reason to have spared Riel would have been his mental state. The fact that the jury asked for his life to be spared would indicate that perhaps he should not have been executed. However, the jury did find that he was mentally capable enough to be found guilty. If he was mentally capable of understanding what he was doing, then he was clearly guilty of treason.
So, your best argument for hanging Riel is the simple fact that he, apparently knowingly, committed treason. This is a crime that was punishable by death at the time (and still is punishable by death in many places).