The pardon is a power that derives from English Common Law as the royal prerogative of mercy, and would’ve been familiar to the Founding Fathers who, in spite of their quarrels with the policy of George III, weren’t quite prepared to rip up their heritage entirely. It can be argued in fact that the Constitution was an attempt to create a British parliamentary system with an American, small-r republican flavor; case in point, note how much attention the Constitution pays to Congress versus the other two branches of government, note the bicameral arrangement and the idea of Senators (the Upper House) being appointed by their State while Representatives (the Lower House, or Commons) are popularly elected.
The office of President, then, can be seen as a desire to make an elected King of sorts—one that possesses vast ceremonial and traditional power but does more reigning than ruling, not unlike the constitutional monarchy that would end up developing in Britain’s political evolution. Among these traditional powers—making war, leading armies, addressing the People of the United States in Congress Assembled, appointing a responsible Government, and conducting international affairs—naturally came the power to issue pardon.