The 1900 Census for Salmon, Lemhi County, Idaho
Peter Amonson, born Dec 1839, 60, born in Norway, immigrated in 1862, naturalized, Shoe Maker, married 27 years (1872/1873)
Cristine(Christina?) Lina(Lene?), (Transcribed as Cristinilina), born Feb 1855 in Norway, immigrated in 1870. mother of 5/5 living
Clara J, Dec 1874 in Idaho
Ester(sic) M, Aug 1883

The 1880 Census for Salmon City, Lemhi County, ID
Peter Amonson, 40, born in Norway, shoemaker
Christina, 27, born in Norway, Keeping House
Clara J, 6
Albert L, 5
Oscar, 2
Anker E. 4/12 (identified as born in Feb)
--All children born in Idaho.

The 1910 Census for Yearianville, Lemhi County, ID
Oscar Amonson, age 32, born in Idaho to both parents born in Norway, farmer/manager, worker.
Peter, age 55, born in Norway, farmer on his own account. widower
--although they were in order on the census, they were identified as being in separate dwelling/separate family listing.
--This indicates Christine died between 1900 and 1910 (But read the 1920).
--The ages of the two are quite different than what they should be. The enumerator might not have understood Peter's English if he had a strong Norwegian accent. Peter may have given the ages, or the enumerator made a mistake.

The 1910 Census for Salmon, Lemhi County, ID, district 203 (page 5 of 29)
Albert C Amonson, 34, born in Idaho to both parents born in Norway, merchant/Gents Clothing, Single

The 1910 Census for Salmon, Lemhi County, ID, District 203 (page 2 of 29)
Esther M. age 27, born in Idaho to both parents born in Norway, clerk/dry goods. Single, lodging on Lane Street in the home of Jessie McBride (female/married) and McBrides two sons, and three other female lodgers.
--As Albert and Esther were enumerated three pages apart, they were not living in the same house)

The 1920 Census for Salmon, Lemhi County, ID
Peter Amonson, 80, born in Norway, immigrated 1865, naturalized 1872
Christina, 60, born in Norway, immigrated 1870, naturalized 1875,

The World War I draft registration card image database:
--Albert Christopher Columbus Amonson, born 27 Feb 1875, Merchant, Amonson & Ayeall(Ayeatt?), married to Ida
Oscar Carleton Amonson, born 1 March 1878
Anker Emele Amonson, born 14 Jan 1880.
--All three very hard to read.

Idaho Marriage Index 1842-1996
Esther M Amonson to William B. Pyeatt, 28 April 1913 in Salmon
Albert C Amonson to Ida M. Simpson, 16 Aug 1916 in Salmon
Emma Webber to Anker Amonson 31 Dec 1903 in Salmon
(From a family tree: Oscar Carleton married Jane Thompson Leach 11 Jan 1908 in New York)

The Idaho Death Index:
Peter Amonson, born 12 June 1839, died 28 Jan 1921 in Salmon, Lemhi Co.
Christina Amonson, born 2 Dec 1849, died 29 Dec 1927 in Salmon
Albert Christopher Amonson, born 27 Feb 1875, died 10 Dec 1948
Oscar Carlton Amonson, born 3 Jan 1878, died 23 July 1942
Anker E. Amonson, born 99/99/9999, died 24 Dec 1929

A family tree of Anker, showed two children, Olaf Edgar Amonson 1906-3 June 1998 in Modesto, CA; and a Karl Amonson 1909-July 1984 in Modesto, CA. Amonson was divorced from Emma, died in Salmon, buried in the Salmon Cemetery.
--I sent a message to this person via the Ancestry contact system, as well as the one above, to read this thread and help if possible.

There is a lot of information available on this family.

As to her family in Norway, have you obtained her parish / emigration records on the Norwegian Digitalarkivet? If you haven't, repost and I can give you some pointers.

Ron Bestrom
Tacoma, WA