Traveling from NZ to USA, how much money do I need for CBP to let me in?

Upon entering the US you will have to prove that you have sufficient funds to sustain yourself for the entire duration of stay in the US. Much to our disappointment, the CBD does not specify how much is "enough funds". There is no fixed amount per day. Indeed, such an amount depends on spending habits, accomodation costs, transportation costs, shopping frenzies, etc. To give you an idea, saying that you'll be staying for a month at a 5* Hilton with 300NZD is likely to raise a few eyebrows.

In addition, you mention that your partner will be hosting you, and funding you, throughout your stay. You should carry a written confirmation of this to be shown to the CBP officer. As the CBP puts it:

Does a foreign visitor have to carry a certain amount of money to enter the U.S.?

Yes. Travelers visiting the U.S. from a foreign country must be able to prove to a CBP Officer that they have sufficient funds (i.e. credit cards, cash, travelers checks, money orders etc.) to cover their travel, lodging, entertainment, meals, etc. in order to be admitted into the U.S.

CBP Officers are aware that there may be circumstances in which a traveler may have limited funds. In those cases they will determine the admissibility of the traveler based on the information provided to support the reason the traveler has insufficient funds.

If you have invited someone to visit you with the understanding that you will be hosting them at your house and providing meals, etc., it is advised that you confirm your invitation in writing so that they have something to show the CBP Officer. The letter should include your full name and address. This will not guarantee their admission into the U.S., but it will help the CBP Officer fully assess their situation.

You are also likely to face extra questioning since you plan on staying for a longish period to stay with your partner. This scenario is often a prequel to overstaying and illegal immigration. You should obviously not lie to the CBP officer. Be honest about your purpose of travel and make sure you have a convincing story about your intentions to leave the US. Showing strong ties to your home country - a job, education career, etc. - usually helps with this. We have a similar question of TSE with tips on the topic: Is it a good idea to mention to the USA immigration officer that I'm visiting my girlfriend?