Aloha 'Oe was written and composed by Hawaii's last last monarch, and only queen regent of the Kingdom of Hawaii, Queen Liliuokalani. After being captured from her home land, she wrote it during her imprisonment, after it was taken over by western industrialization. The reasons behind the song was to express her grievances to that she was unable to protect her people and country, and it was also her way of apologizing for not being able to uphold her duty as Queen.
Its quite possible Nani was made to sing this song due to the fact that, like Queen Liliuokalani who was unable to protect her country, Nani was unable to protect Lilo, and like the queen who had her country taken from her, Nani was going to experience the same with Lilo.
Lyrics that are Sung
Aloha `oe, aloha `oe
Full English Translation
Farewell to thee, farewell to thee
- One of the various meanings to the name Lilo is to "be lost"; which seems to match Lilo in the movie. Especially the scene that shows Nani singing Aloha `Oe to her.