The design of the $5 note was changed in 1990 (fifth issue). The Reserve Bank decided to completely revamp the appearance and features of New Zealand's bank notes. This was the first complete re-design since the introduction of Decimal Currency 23 years earlier. The result, after the Bank had consulted widely with the public, was a new series of notes with distinctly New Zealand designs.
The front of the $5 note shows Sir Ed, Mount Cook and the iconic Ferguson Tractor which was used by Hillary, with minor adjustments, in his trek to the South Pole.
The back shows items associated with Campbell Island (a remote, uninhabited, subantarctic island of New Zealand) - the yellow eyed penguin (hoiho), the subantarctic lily, the Pleurophyllum speciosum (giant daisy) and bull kelp.
So the intent was to show iconic aspects specific to New Zealand rather than celebrate Hillary's ascent of Mt Everest. Given that he was an explorer and mountain climber, including the highest mountain in New Zealand was considered appropriate.
In short - yes - you are right; it was all about giving the design a 'New Zealand touch'.