Second great-grandmother of US Vice President Dick Cheney.
Per the census in 1850, Rachel was born in Tennessee in 1812. This record confirms her relationship to her husband along with the marriage record below, however, it does eliminate Stephen Montgomery as her father as he lived his entire life in Virginia. 
Little is known about Rachel, other than she was the mother of James Alexander Tyler
Her parents are unknown, even professional research done for Lynne Cheney could not identify the parents of Rachel However her marriage record does confirm that her father was a Montgomery.
Land ownership is evinced, but the index record only mentions residence, placing her (or the property) in Franklin, Ohio. A US City Directory entry for Virginia, as well as one other public record in Ohio now shows up on Ancestry.com, partially redacted for non-subscriber accounts.
Rachel Montgomery born 1811 in South Carolina, married Crankfield
Rachel Montgomery born 1811 in Pennsylvania, married Gordon
Rachel Montgomery born 1812 in Kentucky, wife of John Montgomery, also lived in Indiana
Rachel Montgomery born 1810 Virginia, married Phillips
↑ "United States Census, 1850," database with images, FamilySearch (https://familysearch.org/ark:/61903/1:1:MHVS-D94 : 12 April 2016), Rachel Tyler in household of Millett Tyler, Monroe, Pulaski, Indiana, United States; citing family 98, NARA microfilm publication M432 (Washington, D.C.: National Archives and Records Administration, n.d.).
↑ Johnson, Martha Coleman, The Prichard Family: History and Genealogy of the Descendants of James and Elizabeth Hughes Prichard of New Castle, KY, (1915), 28, Heritage Quest (Digital Library).
↑ Cheney, Lynne, Blue Skies, No Fences: A Memoir of Childhood and Family, New York: Threshold Editions (2009), Chart on pages 308-309.
↑ "Kentucky, County Marriages, 1797-1954," database with images, FamilySearch: accessed 12 Oct 2018), Willett Tyler and Rachel Montgomery, 15 Jan 1828; citing Shelby, Kentucky, United States, Madison County Courthouse, Richmond; FHL microfilm 259,285.
Daughters of the American Revolution, DAR Genealogical Research Databases, database online, (http://www.dar.org/ : accessed 12 Oct), "Record of Robert Tyler", Ancestor # A117704.