Based on Maori Language and Culture:
* Te Manu Kai-Ki (The Insect Eater) - "Manu" means bird, "Kai" means food, and "Ki" refers to insects.
* Moa-iti (Little Moa) - A reference to the extinct giant birds of New Zealand, acknowledging the creature's size and possible relation to ancient birds.
* Tūī-nui (Big Tūī) - A play on the name of a common New Zealand bird, emphasizing its size.
Based on Appearance and Behavior:
* Buzzer - Evoking the sound of its insect-catching.
* Hopper - If it jumps and hops like a grasshopper.
* Chirper - If it makes a chirping sound.
* Cricketer - A reference to the insects it eats.
* New Zealand Weta-bird - Combining the familiar New Zealand insect with the creature's bird-like features.
More Creative and Playful:
* Kiwi-chicken - Merging the iconic New Zealand bird with its chicken-like size.
* Insectivore Rex - A fun and slightly ridiculous name.
* The Tiny Terror - Playing on its potential fearsome appetite.
Ultimately, the best name depends on the specific features and characteristics of the insectivore. Consider the creature's appearance, behavior, and your desired tone!