'I am Jesus Gris'

It was his name. Yup, that's it.

Which would be worse, to live as a monster or to die as a good man?

He spends the whole movie becoming something else, but by destroying the Cronos device he regains his humanity. So he's glad that even if he's going to die, he's going to die as himself, not as a monster.

Jesus Gris sounds like Jesus Christ, yes. He is resurrected several times, and in this way reflects Jesus Christ. Also, the literal translation of Gris is gray, so as someone on another thread on this board has suggested, he goes back and forth between black (evil) and white (good) in the course of the film. Obviously, at the outset of the film, he's a good man with love of family (unlike Angel and his uncle, who are both despicable), but when Jesus finds he's made more youthful, vigorous, and attractive to his wife by the cronos device, he's tempted to keep using it. He will not give it up even when he finds himself doing disgusting things like licking blood off the floor. He's giving over to his dark side. It's not until he realizes the thirst for blood the device produces in him is about to make him attack his own beloved granddaughter that he's able to give up the eternal life the cronos machine promises. He then makes the ultimate sacrifice and dies for the good of others, again, the same as Jesus Christ.