Milne has been forced out of the side's upcoming tour of England with an elbow injury and is expected to be unavailable for six weeks.
Bracewell was omitted from the New Zealand Test squad earlier this year for disciplinary reasons.
But New Zealand Cricket's general manager of selection, Bruce Edgar says Bracewell has impressed selectors with his application and commitment over the past three months.
He says Bracewell's worked hard on and off the field to restore his relationship with the selectors and his change of attitude has been noted.
Bracewell, who's 23, has played 18 Tests, his most memorable at Hobart in 2011 when a nine-wicket haul helped New Zealand to a seven-run win over Australia.