Hi MaryAnne In many countries some surnames derivated from the place of origin of the persons. I believe that the name Amyoony refers to belong or come from the Amioun city in the North of Lebanon north east of Batroun, same as Parisian should be for Paris for a person who was surnamed when living in Marseilles or London. If this is right, the first Amyoony was living or working in another place. Also as many lebanese people, emigrated knowing no foreign languages and with turkish passports written in arabic type letters, the american immigration officers just registered them as they could, just guessing and mixing data. So perhaps when they asked his name, he answered the place where he was born or his family came from. I heard of 8 brothers with different surnames. If your grandfather became american citizen, perhaps you can find some more info at the citizenship documents.
I hope my comments can be a little help for you. Best Regards Raul Eduardo Chemmes