Have previously mentioned The Evening Star/Thames star that is online 1874 - 1920 and the Ohinemuri Gazette 1891 - 1921.
Also worth checking are the Waikato Times 1873 - 1886 and Te Aroha News 1883 - 1889.
Early Thames info also appeared in the Daily Southern Cross, the Auckland based newspaper that is online 1843 - 1876. You do find though that all papers in New Zealand did report many evenst relating to Thames and the goldmining activity in particular.
So it pays to start off with a broad search and then if you get too many results, narrow your timeframe, key words, and the newspaper you are searching - from the advanced search page.
You may wonder why I am pointing out the obvious, but I have found several examples recently where a marriage was not reported in the Thames Star but appeared in other newspapers, such as the Waikato Times. Also remember that searching via DIGITAL NZ, allows you to search the newspapers but also a multitude of other resources at one time.