There is no evidence as to exactly how Harold died. It could have been an arrow, in his eye or another part of his body or a sword or knife. The one thing it could not have been was a gunshot as guns had not been invented. The Bayeux Tapestry was made some years later under orders from supporters of William the conqueror, and tells the story of the run up to the Norman invasion and the battle. There is a scene in the tapestry showing someone with an arrow in his eye but it is not clear that this is Harold, and this is probably where the idea came from. The arrow through the eye might be a Norman invention but it is logical. The normans and Harold were christians and blinding was said to be a punishment from god for breaking a promise. Edward had made a promise to William that he could have the throne. There was also a few other rumours but I don't think anyone will ever find out unless the Human Race finally perfect time travel!