Henry Davis would die in office on August 22, 1889. Samuel W. Chubbuck was appointed to fill the vacancy on September 9, 1889, but resigned on November 30, 1889. The next appointee was Frank Bell, who took office immediately on November 30, 1889. Bell would subsequently become acting governor on September 1, 1890 when the incumbent governor, Charles Stevenson, ceded his powers due to disability. Stevenson died on September 21, 1890. As Bell only served as Acting Governor, even after Stevenson's death, the office of Lieutenant Governor was technically not vacant for the remainder of the term.
The primary source is the original manuscript canvass, located at the Nevada State Archives in Carson City. The results were also printed in the 1885-1886 Biennial Report of the Secretary of State.
Source for Popular Vote data: Nevada State Archives, "Lieutenant Governor," Canvass of Votes Cast at the General Election held in the State of Nevada on the 2d day of November, A.D. 1886 (Carson City, 1886)