SEOUL—Samsung Electronics Co. nominated Lee Jae-yong, the son of its longtime chairman, to its board of directors as shares in the technology titan suffered their steepest one-day decline since 2008 in the wake of a global phone recall.

The latest step in the elevation of Mr. Lee, the South Korean company’s vice chairman and heir apparent, comes more than two years after his father, Chairman Lee Kun-hee, was hospitalized following a heart attack. He remains incapacitated.

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