I do not yet know much about them - more research needs to be done. What I found, I found on the LDS web site, Family Search... or, at least, what I initially found.
I found the birth and baptism information for Ole Olsen Melom in Bruflat i Sør Aurdal 1866-1893 on page 29, line 6. He was born 2 January 1874 in Sør Aurdal and baptized at Bruflat Kirke on 1 March 1874. His older cousin, Knud Olsen Fjeld (first-born son of my immigrant ancestor, Ingri Melom) was one of the witnesses at his baptism.
In addition, I found Ole Olsen Melom in the 1910 U.S. Census living in Herr, Sheridan, North Dakota. According to that census, he immigrated to the United States in 1883. Ole would have been about nine years old in 1883.
By contrast, the 1910 and 1900 census data for his father, the elder Ole, indicate the family immigrated in 1879. Thinking to resolve these discrepancies once and for all, I went to the digitalarkivet and checked up on the entries for those leaving the Norway in the parish records. Then, I went to the "Digital Inn" and perused the emigrantprotokoller, etc. I looked under every year available in all the records and could not find this family anywhere. However, it must be noted that over 800,000 Norwegians immigrated to the United States between 1825 and 1939. The Digital Archives of Norway has 700,000 of these people digitized. So, perhaps, the Meloms are among the 100,000 plus individuals who left Norway and do not appear in the digital records?