I searched the two books on Thurston's line. The books were compiled by Dr. Stuart P. Howell, Raynor Family genealogist Thuston lineage. They are Josiah Raynor of Manorville of Long Island, New York in which there are quite a few Lane surnames listed. But none match your Andrew. Also I searched The Raynos of Ketchaponack which had a few Lanes listed. Possibly you could send a email to the Raynor Family Association at email@example.com. Hopefully they can assist more than I can.