Fanny Lines was born in 1789 in New Jersey. Her father may have been Anthony Lines of the German Lines, or Leyn, family which resided in the Pompton Lakes region. Fanny and John Simonson had 5 children: Barnabus, Jane, Anthony, Isaac and Caroline. Fanny died December 7, 1869. She is buried in the Caldwell Presbyterian Churchyard. Her findagrave.com entry is here.
My support for the idea that her father was Anthony Lines comes from this document seeking pension recompensation for his surviving child Fanny Simonson, who in the document is referred to as Barney [Simonson’s] wife, but who was really John [Simonson’s] wife (Barney was her father-in-law). But her two sons are named correctly.
Selected Final Pension Payment Vouchers 1818-1864
Lines, Anthony, N.J. 1832
Passaic Co., N.J., 23 Sep 1840, George A. Ryerson [s], Clerk of ICCP, certifies court satisfied Anthony Lines was a pensioner (former private in Rev.) who died on 25 Jul 1840 leaving no widow but one surviving child only: Fanny Simonson (widow of Barney Simonson). On same date, Fanny Simonson [s], child of the pensioner, appoints John Titus atty to collect arrears due from 4 Mar 1840 to 25 Jul 1840; ack. before Andrew Mead J.P. [s]; witness Anthony Simonson [s] and Isaac Simonson [s].
Trenton, 24 Sep 1840, John Titus [s] collected $31.31; his oath heard same date and place by Mayor Chas. Burrough [s].
This clue seems to point to Anthony Lines being Fanny Lines’ father. Her husband, John Simonson had died in 1828. John Simonson’s father, Barney, was also a Revolutionary War soldier. Fanny’s sons Anthony and Isaac would be the right age to witness (ages approx. 30 and 29).
The Olson-Sumner Ensemble 
Anthony Lines (6 Apr 1758 – 25 Jul 1840) was married to Elizabeth Moore and lived in Acquackanonk Township in Essex County (later Passaic County). She was the daughter of Benjamin and Martha Moore of Essex County who migrated after the Revolutionary War to Albemarle County, Virginia.
See information about Anthony Lines at http://www.werelate.org/wiki/Person:Anthony_Lines_(1)
John Everett Jones and I have had some email conversations about this. He now thinks the Anthony Lines who married Elizabeth Moore was not this Anthony Lines.