Black Caps selectors said they believed a broken thumb Williamson sustained in Bangladesh in October will be healed enough for him to take the field for the First Test in Dunedin on Tuesday.
"Kane has been in the nets batting and bowling and getting his intensity levels up," selection manager Bruce Edgar said. "We're confident he should be right come Tuesday."
But Edgar said Aaron Redmond, who has not played a Test for five years, was included in the squad as cover in case the gamble did not pay off and Williamson was unable to play.
Sodhi replaces off-spinner Bruce Martin, who has failed to fire since earning his first Test cap last year at the age of 32.
Veteran spinner Dan Vettori ruled himself out of the squad this week as he battles a long-term Achilles problem and troubled batsman Jesse Ryder's hopes of an international return stalled last week when he was injured in a domestic match.
New Zealand: Brendon McCullum (captain), Corey Anderson, Trent Boult, Doug Bracewell, Peter Fulton, Aaron Redmond, Hamish Rutherford, Ish Sodhi, Tim Southee, Ross Taylor, Neil Wagner, BJ Watling, Kane Williamson.