President-elect Donald Trump on Friday selected Donald McGahn, his chief campaign attorney, to join him in the White House as his top lawyer, sliding the longtime election-law expert into what promises to be a high-profile job with unique challenges in the next administration.

Mr. McGahn, 48 years old, has been a partner since 2013 at the Jones Day law firm, where he has focused mostly on political​ law. He first met with Mr. Trump in 2014 as the New York businessman was exploring a potential presidential campaign.

“Don...