I used the internet cafe at this place this past spring:

Waikiki Beachside Hostel
2556 Lemon Rd.
Honolulu HI, 96815

The rate was cheap, I forget how much.
The connection speed was OK.
The surroundings and atmosphere were moderately scuzzy, located in a seedy lane and a lot of backpacker types around.
It is quite near the zoo if that is what you are looking for.
I will choose a different place next time if I can find one.