Question regarding storing luggage, hostel or airport?

I'm planning a trip in February next year and I want to visit Cook Islands for 5-6 days on my way back from New Zealand.
I want to go to Aitutaki as soon as I get to Cook Islands and then spend a couple of days on Rarotonga before I fly home. I want to do this because I don't want to risk missing my flight to Los Angeles if I get stuck on Aitu, and that is why I plan to do the Aitu trip first and then give myself two days on Raro.

The flight from Auckland gets in at 2:45 in the morning and the first flight to Aitu leaves at 8 in the morning. Is there a place at the airport where I can store the luggage I cant bring to Aitu? Or can I leave it at a hostel for a couple of days? (I'd most likely stay at that hostel for a few days when I get back to Raro)

There are lockers for luggage storage at the airport on Raro. You go to the security office and pay a few bucks and they'll give you a padlock. The hotels are generally helpful with keeping bags, early and late checkouts, too, so check with the hostel you have in mind to be sure.

The Aquarius Rarotonga 'boutique' hostel is 2 mins (literally) walk from Raro's airport terminal, right on the seafront. I think it has 24hour reception, the dorm room is very good quality indeed, and spacious. You could stay there, get a few hours sleep, a wash and breakfast before heading back to catch your flight (you can even go do your flight check in and deposit your hold luggage then go back to the Aquarius for a lazy breakfast and stroll back to the terminal shortly before boarding time). You should be able to negotiate a discount rate for the limited time you are there, staff are friendly. All hostels on Raro will lock away bags for you whilst you visit Aitutaki, they all charge a small fee for this.

The Paradise Inn in town (Avarua) will usually store your bags free of charge if you are planning to stay there on your return from Aitutaki, and most other places also do not charge for this service. E-mail the ones you are interested in. But Aquarius is a good choice as well, as indicated above.