Young, 28, spent last season with the Philadelphia 76ers, averaging 10.6 points and 23.9 minutes per game.
"At 6-7, Nick's size, ability to create his own shot and athleticism make him a versatile player who will give our lineup multiple looks on the floor," said general manager Mitch Kupchak in a statement. "He's an exciting player, and we're excited to have him on our roster."
The free-agent guard signed a one-year deal at the veteran's minimum with the Lakers, according to USA Today.
"It's great for Nick and great for the Lakers," Young's agent, Mark Bartelstein, told the Los Angeles Times. "He could've waited and got more money in the free-agent market, but his dream has always been to play for the Lakers."