The 27-year-old, who has 46 Scotland caps and can also play in the second row, has effectively replaced Richard Hill in the squad.
Hill damaged knee ligaments in Saturday's first Test defeat and is out of the tour.
White was considered unfortunate by some not to have made the initial Lions party.
He was recovering from broken thumb during the early weeks of the tour and the Lions opted for Irish blind-side flanker Simon Easterby to replace the injured Lawrence Dallaglio.
He said: "I'm not sure if I'll get any game time because obviously it's late on in the tour by the time I arrive but if I do I will just play my own game and do my best to help the rest of the guys out there.
"I was disappointed not to make the (original) squad. I had my holiday in Scotland, did training every couple of days and kept my fitness ticking over. I'm just delighted to be called up."
White becomes the 51st player called up by the Lions and the sixth addition to the original squad.
Iain Balshaw was replaced by Mark Cueto before the party even left for New Zealand, since when Malcolm O'Kelly and Simon Taylor were invalided home without playing a game.
Lawrence Dallaglio fractured his ankle 21 minutes into the opening game against Bay of Plenty, and then came the Lions' nightmare Saturday.
Not only did they lose the first Test but captain Brian O'Driscoll dislocated his shoulder and Hill suffered knee ligament damage for the second time in eight months.
In addition to that, Welsh centre Tom Shanklin is off the tour with a knee problem while lock Danny Grewcock is to return home after being suspended for two months for biting All Blacks hooker Keven Mealamu.
Grewcock was replaced in the squad by Welsh lock Brent Cockbain and White now arrives as Lion 51, though he is unlikely to be available to play until the final midweek game, against Auckland on July 5.
The loss of Taylor to a hamstring strain had reduced the number of Scottish Lions to just two - hooker Gordon Bulloch and scrum-half Chris Cusiter.
Neither were involved in the first Test but both will play against Manawatu, with Bulloch captaining the side for a second time.