It means Trent Boult misses out, along with Tim Southee, who was left out of the 14-man squad for the second test.
Both Arnel and Gillespie have been in good form in the domestic competition but it lengthens an already long tail.
South Africa have won the toss and have put New Zealand into bat.
South Africa will field the same XI who played in the first test in Dunedin, with AB de Villiers' ankle injury not serious enough to keep him out.
The day has dawned overcast but fine in Hamilton, raising the prospect of the team winning the toss bowling first.
Teams:
New Zealand: Ross Taylor (c), Martin Guptill, Rob Nicol, Brendon McCullum, Kane Williamson, Dan Vettori, Kruger van Wyk, Doug Bracewell, Brent Arnel, Mark Gillespie, Chris Martin.
South Africa: Graeme Smith (c), Alviro Petersen, Hashim Amla, Jacques Kallis, AB de Villiers, Jacques Rudolph, Mark Boucher, Vernon Philander, Dale Steyn, Morne Morkel, Imran Tahir.
Umpires: Richard Kettleborough (England), Billy Doctrove (West Indies). Third umpire: Aleem Dar (Pakistan).
- Michael Brown
