For example in Persian poetry the word liver is found pretty much everywhere you would expect to see the word heart.

"You are my liver" is a nice/romantic thing to say to this day.

It could have been because the liver was regenerative and they were using the titan with the regenerating liver as another reference to humanity, or it could have been as someone pointed out because predators like to eat the liver first, or it could have been that you can take damage to the liver and live for hours afterwards in agony whereas the same amount of damage to the heart or lungs is apt to cause death to come much quicker.

Perhaps there are undiscovered variants out there were the eagle or a serpent or some other animal attacks his genitals, or his eyes.

But at any rate the Eagle ate his liver, it means nothing in particular, and at the same time it means every particular interpretation we can place on it.