Harold B. Lee was relatively young when he took over as President of the LDS Church at age 73. Most people thought he'd be in charge for the next 15-20 years, but he was in charge for less than 18 months before his sudden death. Therefore he only had the opportunity to call one Apostle.
(July 29, 1915 - April 19, 1985)
Apostle - October 12, 1972
McConkie was born in Ann Arbor, Michigan. He served his mission in the Eastern States and married Amelia Smith, daughter of Joseph Fielding Smith, while the two fo them were at the University of Utah. He was a WWII vet, working for military intelligence.
After the war he worked as a city attorney and as a reporter for the Deseret News. He was called to be a member of the First Quorum of the Seventy at age 31 by Pres. George Albert Smith.
In 1958 he wrote Mormon Doctrine, which tried to serve as a one-volume encyclopedia about the doctrines of the LDS church. Many of the Brethren were unhappy with its publication, since it wasn't authorized and had some errors and an overall harsh tone. The 1996 version had some revisions but it's since gone out of print.
When McConkie was called to be an Apostle after the death of his father-in-law, he said he'd known for some time that this calling would come. He was seen as one of the more authoritative voices in the Q12. He was quoted in 1980 saying to an LDS scholar: "It is my province to teach to the Church what the doctrine is. It is your province to echo what I say or to remain silent."
His final testimony at the April 1985 General Conference, when he knew he was dying from cancer, was a powerful one about the Savior. He died less than two weeks later.