Check the following "Settings". Make sure under "General > Date & Time", that your iPhone is showing the correct "Time Zone" You can either have the iPhone determine this automatically (by turning on the "Set Automatically" switch) or you can set it for a specific location. Do note that in order for this to set itself automatically you need to have location services enables as well.
The next thing I would check is "Settings > Mail, Contacts, Calendar" and make sure in the "Calendar" section that you've enabled "Time Zone Support".
All of this is covered in this Appple knowledge base article: http://support.apple.com/kb/ht4576
Hope this helps. Good luck!
Following some trial and error, and additional reasoning, I can partially answer my own question:
(1) The events from a local calendar, on an iPhone ("On This Phone", I think it is referred to), are not backed up to the calendar-specfic area in iCloud. Here:
iCloud.com > Settings > (Advanced section) > Restore Calendars and Reminders
Although it doesn't say so explicitly, it makes sense that these iCloud calendar backups cover a user's iCloud calendars. (Plus, my attempt to restore such backups with a hope to see pre-merged, local calendars, failed).
Screen shot from iCloud.com:
(2) The iPhone backup (the complete backup of the whole phone), which in my case was being backed up to iCloud (rather than iTunes), would, I presume, include the local calendars and their events.
However, by time I'd got to looking at this, my oldest 'complete' iCloud backup was just a day old... which was after I merged the local calendars into iCloud calendars.
Best Answer
If you set up the events using Hawaii times (ie you set your Hawaii noon meeting as an 8pm meeting while editing the calendar in New York), then there is no problem since when you set your device to Hawaii time all those events will be correctly displayed.
If you set up the events using NY times (you set your Hawaii noon meeting as a noon meeting while editing the calendar in New York), then you need to either edit all the Hawaii meetings or tell the calendar to ignore the time zone info and just display the events as entered by using "Time Zone Override" on the iPhone and setting the iPhone calendar to display events using the New York time zone.
To turn on Time Zone Override:
Go to Settings > Calendar
Move the switch to show green in order to turn on Time Zone Override.
After turning on the Time Zone Override, you need to select the New York time zone.