There are 2 cheap options for parking nearby the Honolulu Cruise Ship Port. The first one is Mark's Garage on Bethel Street. Here you can pay a flat rate of $10 for all-day parking or just $5 if you arrive after 3:00 p.m. The second option is the parking lot on North Beretannia Street and Aala Street, where you can park for a consecutive 12-hour period for just $4. Please keep in mind that the first option doesn't allow overnight parking, while the second option does offer overnight parking.
For more info, see our feature Honolulu Port FAQ