In two previous posts on Sedeney Tilley, wife of Samuel Wallis and her family, I've written about this mysterious family. Mystery solved. Thanks to T.J. Rand and Google Books, I was pointed toward the Lary Family Genealogy. This is only a snippet view in Google Books, but the actual book (you remember books--they are made of paper and live in a library) was easily found and read. The Tilley family has its own brief chapter in this book on pages 51 and 52. They seem to be an Irish family that emigrated to work mining bog iron. Samuel Tilley's name is listed on the "Irish Lotts" in a 1729 map of Barrington. Samuel and Jane (---) Tilley were the immigrants. They had two daughters. One was Jane, the wife of Daniel Lary (which is a variant of Leary or O'Leary, another Irish immigrant). She sold her half of her father's land in April 1773 and moved to Somersworth where her five children are given in Master Tate's diary. She named her youngest daughter Sidney. The other daughter is our Sedeney, which is incredibly enough, probably Sidney too. I can accept that name because we are now not talking about colonial Puritans in the 1720s to 1740s, but Irish families. Of course, she sells her half of the land in 1793.
The full citation: Carleton Edward Fisher, Lary Family Genealogy: Daniel Lary of New Hampshire and His Descendants (Rockland, Me.: Courier-Gazette, 1977).