I'm not related, but wanted to see if I could help. I'm afraid that what I found may raise more questions than it answers.
In 1900, Botan is listed in the Benj. F. Brooks household following daughter Rachel & a son-in-law named Fred - both listed under the surname "Brooks" which is probably incorrect. The record indicates that Rachel, who is 21 years old (she was 1-year-old Rachel D. in the 1880 census), married for 3 years, and is the mother of 3 children with 3 living. The son-in-law, Fred may actually have been Fred Akers. I found a marriage record for Rachael Brooks and Fred Akers March 6, 1897 in Iron County.
In the 1910 census 14 year old Botan is living with 67 year old widow Elizabeth Brooks. He is listed as being her GRANDSON. My guess is that Botan was born to Rachel before her marriage to Fred - thus he had the surname Brooks.
In the 1900 census, Elizabeth's initial looks more like an "N" to me, rather than a "U" which it is indexed as at Ancestry.com & FamilySearch.org. I've seen her listed in a couple of online family trees as "Nancy Elizabeth".
There is a marriage record in Iron Canyon, Missouri, July 25, 1861 between a Benjamin F. Brooks and an Elizabeth Thompson.