Mary Ann Craddock born Jan 1819 in Tennessee, found in 1870 census in Holden Twsp Johnson County Missouri. Widowed keeping house with another widow by name of Nancy Willderman.
Meanwhile your Luther Martin Green is found in Johnson County with the Lively family in 1880 working as a farmhand and listed as illiterate. Appears he married "up" when he married Lucinda Huey as she and a number of her relatives apparently moved to Texas and then Oklahoma between 1873 and 1878. Her father Thomas V. Huey was a private in Co A 41st Mississippi Infantry and was at Stones River and Chicamauga FWIW. Not sure how Luther Martin Green came to be in Oklahoma as the marriage to Lucinda is listed as occuring in Oklahoma in 1887, unless he was a part of the land claim runs.
Furthermore,
Luther, spelled Leuther in the indexes, is found as age 8 in the John Rosencrans household in Rose Hill Twsp, Johnson County Missouri in 1870.
This census story would indicate that Mary Ann (Craddock) Green lost her husband between 1862 and 1870 in or around Pettis and its adjoining county of Johnson. This certainly raises the spectre of a Civil War death.
Thus perhaps the best place to start would be looking at Pettis or Johnson county probate and gurdianship records for the period. If fatherless and farmed out to foster care at age 8, it certainly is undestandlable how Luther Martin Green may not have had much knowledge of his father.
There is a tantalizing bit of history via the History of Adair Co by Violette pg 103 that lists a Reuben Green being executed after the Battle of Kirksville (pvt, confederate no unit given) on August 7th 1862. Not sure if it fits and no proof he is your guy but is the right time frame. The Brig Gen Martin E Green of Lewis Co who was killed at Vicksburg, though sharing the name Martin, is not your fellow.
Have fun...
John R.