I found your mother in the Verheecke Family Tree, trees.ancestry.com/tree/21789313/person/1134320192. You probably want to contact the owner. That family tree says that Mary Henry Verheecke died 30 Sept. 1993 in Rapid City, Pennington, South Dakota.
You'll want a copy of her obituary because that should list all of her siblings and where they were living in 1993 and might mention her parents' names.
I don't know about finding obituaries in South Dakota but that's a good question to ask on the Pennington County Message Board.
I also found a Lucy G. Nicolai
Born: 10 Oct 1918
Last Residence: Kwethluk, Bethel, Alaska
Died: 17 Sept. 1996
Source: Social Security Death Index.
That Last Residence field may NOT be correct because the Social Security dept. uses it for other info too. However since it's the only clue we have at this time, might as well start there. We can't look at her death certificate because they're on hold for 50 years from the date of death.
Lucy's obituary might be found at the Alaska State Historical Library, http://library.state.ak.us/hist/newspaper/newspaper.html
You'll want to look into the records at the Alaska State Archives, archives.state.ak.us/genealogy/genealogy.html .