After the third day's opening session was lost to rain in Hamilton, Williamson found his methodical groove to advance to 123 not out.
Daryl Mitchell was alongside him on 18 as the home side reached 274-4 at tea in their second innings at Seddon Park.
Rachin Ravindra was the only wicket to fall, caught at mid-off off Matthew Potts (1-53) for 44 after resuming on two.
New Zealand batted sensibly to add 138 runs from 36 overs with Williamson -- who resumed on 50 -- posting a 33rd career century on a ground where he historically thrives.
England's struggles under gloomy skies were exacerbated by an apparent hamstring injury suffered by talisman Stokes in his third over of the day.
The influential skipper immediately limped off.
Stokes suffered a torn hamstring on the same left leg in August, forcing the 33-year-old to miss four Tests during a two-month recuperation period.
He returned to action midway through the recent Test series in Pakistan.
Williamson had struck 16 fours and one six but rode some luck on his way to three figures.
New Zealand's greatest run-scorer survived a tight Brydon Carse lbw appeal review on 73 and was later dropped by wicketkeeper Ollie Pope, when a difficult leg-side chance spilled from his glove off Stokes.
It allowed Williamson to score his fifth century in successive Tests at his home ground in Hamilton, where his career record is remarkable.
In 21 innings he boasts an average of 98.81.
His 1,581 runs at Seddon Park is now the most by any batter at any New Zealand Test venue, surpassing the previous record held by Williamson himself, at Wellington's Basin Reserve (1,538 runs).
Overnight rain had continued into the morning, forcing match officials to abandon the first session.
Play started two and a half hours later than scheduled once the outfield had dried.
The forecast for the remainder of the Test is favourable.
New Zealand resumed on 136-3, holding a 340-run advantage after the tourists were skittled for 143 on Sunday.
England have already wrapped up the three-match series, leading 2-0 after emphatic wins in Christchurch and Wellington.
