I am a direct descendant of Robert Huston Lute and visited Hot Springs, Fall River Co, SD, several years ago trying to find the answer to this question. I was fortunate to be able to talk with the woman who was running the historical society (she has since died) who was of the opinion that the grave no longer exists and that he may well have been buried on the homestead. He did die from a fall off a house roof -- probably while repairing or building it.
Interestingly, his widow Sarah sold the land less than a year after his death. Upon the sale, it was discovered that the grant had never been recorded (if in fact Robert had applied for one). Sarah had to apply for it to perfect (i.e., make good) the deed when she transferred the property. On that grant she was called Sarah Lute of Stewart. Stewart was the early name for Minnekahta which is now only a highway intersection and has no post office although the people of the area still refer to that part of the county as Minnekahta.
It is a flat, hill-rimmed area on which grains were raised and Livestock were kept. Minnekahta is an Indian work maening "hot water" (don't know which language). The junction is between Edgemont and Hot Springs. The land is at the NE intersection of Hwy. 89 (on the west) and Hwy. 18 (on the south), about 12 miles west of Hot Springs, comprising 160 acres for which Sarah paid $200 at $1.25/acre. She sold it for $850.
E-mail me direct if you would like further information.