Harriet Beecher Stowe's well received novel Uncle Tom's Cabin (1852) was obviously inspirational or at least explains this song cluster's success.
Ted Snow [former slave looking back on the years he spent in some Uncle Tom's Cabin]
Mr. Knoll & Miss McNeal [cornet duet as Old Log Cabin In The Lane]
Metcalf & Spencer [idem]
Bentley Ball [as De Little Old Log Cabin In De Lane]
Fiddlin' John Carson & Moonshine Kate [Ralph Peer recording for Okeh]
Carter Family [as Little Log Cabin By The Sea]
Fiddlin' John Carson [as Little Old Log Cabin By The Stream]
Cal Turner [idem]
Bill Monroe [as Little Cabin Home On The Hill; not crediting Will S. Hays, probably because the copyright had expired (over 70 years)]
Million Dollar Quartet [as Little Cabin On The Hill; they also cut Bill Monroe's Summertime Has Passed And Gone and Sweetheart You Done Me Wrong ('48)]
Incredible String Band [as Log Cabin Home In The Sky]
Elvis Presley [as Little Cabin On The Hill on Elvis Country, crediting Bill Monroe & Lester Flatt]
Parodies: Another Fall Of Rain, Little Joe The Wrangler, Little Old Mud Cabin On The Hill, The Double Breasted Mansion On The Square, Little Red Caboose Behind The Train. The melody also served gospel hymn Lily Of The Valley (1881), written by Salvation Army soldier Charles W. Fry, first recorded by Welling & McGhee (Vocalion - '28), also cut by Judy Collins, Eddy Arnold, Forrester Sisters, Johnny Bond and Burl Ives. Maybe it's the other way around, Lily Of The Valley as inspiration for the Little Old Cabins.