Spoilers haha. When Rapunzel's mother was pregnant with her, she became really sick. So a bunch of people searched for the magic golden flower that had healing powers from the sun so they could make the queen well again. It was successful, so the queen was healthy and gave birth to Rapunzel, who got the powers of the magic flower in her hair. When Eugene was dying and Rapunzel had her hair cut off, she tried to sing her healing song, but it didn't work, until one of Rapunzel's tears fell on him and healed him and brought him back to life. So basically the answer is... Disney Magic! :)

^ Adding onto that, it harkens back to the original tale where the Prince became blind and one of Rapunzel's tears was able to heal him, and restore his sight. In the Disney tale, the magic is tied to Rapunzel's hair and once it's cut, the magic is lost forever. So, after all Rapunzel's hair was cut off, her tear was the last drop of magic she had, and that's why it was able to heal Eugene. It's also hypothesized that that tear drop was the original drop from the sun that created the golden flower. I

This should've been in the answers section.

It's the tear.