Of course Smith is a very common name and very difficult to research. Your best source would be the 1880 U.S. Census in which I would look for Mike Smith born 1878. Finding a death record for their mother also would be usefull.
These web sites may be of help:http://www.linkpendium.com/https://beta.familysearch.org/
“LDS Family Search” If you place your browser on the name on the left, more information comes up. http://www.findagrave.com/ http://www.deathindexes.com/ http://search.ancestry.com/search/db.aspx?dbid=70734
“Billion Graves.com Burial Index”
Census records can be searched on-line for free through Heritage Quest. This research tool is available through most major libraries if you have a library card with them. The link can usually be found under the library’s web page usually under the category “Research Resources or Tools - Databases”.
There was coal mining in eastern Oklahoma and western Missouri. If you can come up with a more specific Smith name, I would be glad to search my records. I have many mining accident records that could fit your criteria.
“Mining Accidents U.S., Canada, New Zealand and Australia”