Source:1841 Oregon Territory Census: Gingras, Jean Gingras.
Source: 1845 Oregon Territory Census: Gingras, Jean. In Champoeg (now Marion Co.)
Source: US Federal Land Claim, OREGON: Jean Gingras 17 Oct 1845 in Champoeg.
Source: US Federal Census:
Jan 1850 Marion Co. Oregon Territory: Gingras, Jean, 49, (b. 1802 Yamaska, PQ.) w.Olive, 16, Oregon Territory. s.Joseph Narcisse, 18, Oregon Ter. d.Lizette, 9, (ab.1841) in Ore. Ter. s.Louis, 3, Ore. Ter. s.Francois, 1, Ore. Ter.
I think that the daughter Lizette might be your Elizabeth which could be an English form of the French name of Lizette.Olive was Jean Baptiste Gingras 3rd wife, she's not Lizette/Elizabeth's mother.Jean was married first to Anneas PAUL, around 1825 in Canada. Then he married, Charlotte "Striped Dress" SKEALKS or OKANAGAN on 20 Aug 1841 in Okanangan Co. Washington Territory.Then he married Olive FORCIER on 28 Jul 1845 in St. Paul, Marion Co. Oregon. So Lizette/Elizabeth's mother is probably Charlotte, a Native American Indian Woman of the Okanagan Co, Washington area.
Jean Baptiste parents were Joseph GINGRAS & Marguerite BELIVEAU of St. Michel, Yamaska Co. Quebec Canada.