She finished the first day at -3 and was only one stroke behind the leaders, the American Megha Ganne and the English Mel Reid.
Henderson shares third place with two other golfers, Americans Megan Khang and Angel Yin.
The Canadian was in the first row after 17 holes, but she put three putts on the 18th pennant to slip to -3 on the course of the Olympia club.
The day, however, was Ganne's business. She handed out a 67 (-4) card to become the first amateur golfer in 15 years to share the lead in this major tournament.
The 17-year-old streaked birdies in the second nine to grab the top spot on her own, but she bugged the 18th hole to end the day atop the standings with Mel Reid.
Ganne became the first amateur golfer to share the lead of this tournament since Jane Park in 2006.
She needed a win in overtime last month to qualify for the US Open for a second time. Two years ago, she could not avoid the ax.
Ganne had a birdie on three of his first eight holes and added three more on the back nine. She committed two bogeys.
Defending champion South Korea A Lim Kim struggled a lot and signed a 79 card.
Quebecer Noémie Paré and Ontarian Rebecca Lee-Bentham also take part in this tournament.
Paré ended the day at +7.
Collin Morikawa ahead in men's
Collin Morikawa collected seven birdies, including three in a row, to lead with one stroke at the Memorial Tournament.
Adam Long is just one stroke behind, while Canadian Nick Taylor is one of four at 68. He has six birdies.
Taylor had a great first nine, with three birdies and an eagle on the fifth hole.
Bo Hoag, Xander Sc Chaudele and Rafa Cabrera Bello also played 68.
In a large group at 69 are Justin Thomas, Rickie Fowler, Shane Lowry and Jon Rahm, who is the defending champion.
Canadians Corey Conners and Adam Hadwin handed in 74. Conners recorded a double bugy, as well as three bugys in the span of four holes. Hadwin was also the victim of a double bug, in addition to committing five bogueys.
Action will resume early Friday morning.