Its likely that Rhaegar and Lyanna ran off and got married because we know that Lyanna didn't want to marry Robert Baratheon. As she knew he was a whoremonger, and Rhaegar was just a romantic harp playing prince.
Why Rhaegar, who supposedly was gathering people to depose his crazy ass dad, would run off and marry someone already betrothed while at the same time he was already married is the real question. By all accounts Rhaegar was no fool. He must have known something would happen because of his rash actions.
I guess he just didn't expect his crazy ass dad to burn the Warden of the North and his heir alive in the Red Keep then call for the heads of all the Starks and Arryns as well? Thus causing the war that ended his line.
