What is the ruling on naming with "Lina" and "Naila," and is it true that the name "Naila" refers to an idol? And what should be done if that is the case, knowing that the daughter is three years old and the family lives abroad?

There is no objection to naming a girl "Nailah"; it is a name shared by many female companions (Sahabiyat), and the Prophet, peace and blessings be upon him, did not disapprove of this name or order its change. The fact that it was once the name of an idol does not prevent its use, for the general rule concerning names is permissibility unless prohibited by the Sacred Law. Its linguistic meaning is "gift" or "bounty." The Prophet, peace and blessings be upon him, did not change the names of the female companions who bore this name, and none of the companions or early scholars objected to its use. The difference between "Nailah" and other names of idols is that "Nailah" was originally a woman who was transformed into a stone, and then worshipped, whereas other names are proper nouns specifically for idols.