This is a walkthrough on searching tips and tricks for the www.familysearch.org site, which is the online database from the LDS records. The link to the site is at the bottom of this article.
Firstly, it must be understood that not all records in this site are accurate, there are typos, wrong dates, multiple records for the same person with different details (for example, different centuries for birth dates.)
However, this site is good for getting information, but I strongly advise that you confirm the information you get from this site through another source.
NOTE: My only experience with this site is for Maori names only, although I am sure that most of what I say here applies for pakeha names.
1. Go to the familysearch site, then click on the search tab.
2. Put a surname in the Last Name box
3. Set the country to New Zealand
4. Click search
If you are looking for a surname like Te Ingoa, leave the Te out and just put in the Ingoa.
If you are looking for Maori who only have one name as they were born in the 1700's, put that name in the Last name box.
International Genealogical Index - Southwest Pacific is the main place where you want to look as this is the records for new Zealand. There may also be records under Ancestral Files as well as Pedigree Resource Files which may be well worth a look. don't worry about any of the US results that may show, unless you have whanau in the US.
If you find that you have many results, scroll through and see if the places ring a bell, for example, if you whanau is from Whangarei, and some of the results have the place as Invercargill, then the chances that these are your whanau could be slim.
Have a look at all of the relevant records, even if you do not know the people, for you may find the connections.
Make sure you have read the information on "Maori Naming Conventions" as this may also guide you to find people under different names.