Williamson suffered a left pectoral muscle injury while fielding in the second test in Wellington, which New Zealand won by 12 runs to take an unassailable 2-0 lead into the final match which starts on Saturday.
"Kane is obviously a really important player for us and with the nature of the injury we couldn't be sure he wouldn't aggravate it," said coach Gary Stead in a news release.
"He's improving every day and will continue his rehabilitation as planned. Will has been waiting in the wings for a while now and he'll get a great opportunity batting at number three in a settled line-up."
There was better news about wicketkeeper BJ Watling, who passed a fitness test on a hamstring strain on Friday and will play at Hagley Oval.
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Pace bowler Tim Southee will captain the side for the first time in Williamson's absence.
(Reporting by Nick Mulvenney in Sydney, editing by Peter Rutherford)
