Sorry for the delay in a response to your post.
Essentially there are very few (if any) rental car companies that operate from paihia.
Kerikeri is the main hub for the bay of Islands in terms of rental cars, as the airport for the Bay of Islands is in Kerikeri.
So you could easily rent a car from Avis or Hertz from Auckland, and drop off in Kerikeri.
Kerikeri is a great town to stay in, I would personally prefer it in a lot of ways to Paihia, so don't know if there is any specific reason you wanted to end up in Paihia - maybe you could let us know a bit more about your plans once you are in the bay of islands and I could give you some more advice.
Also, just as another possibility, there is a company called 'Go Birdz' which has rental locations in Auckland and Kerikeri, but will also arrange to pick up/drop off at Paihia locations; they rent more budget vehicles than Avis or Hertz, so they may suit your needs.
I have never dealt with this company, so can't give any personal recommendation - just know that they exist....