Official comment
This response has been edited to clarify behavior with the various options in Campaigns.
If the Wait is set for a 1 day delay in the Contact's time zone, without a specific time for delivery set, the Wait will hold the Contact for 24 hours and then wait until 10 AM in the Contact's time zone comes around. For example a Contact that lands on the Wait at 10:01 AM on Monday will wait until 10:01 on Tuesday for the "one day" wait, and then wait until 10:00 AM on Wednesday to proceed out of the Wait.
The same methodology is used for a specific time set; a 1 day delay is 24 hours, then the Wait holds until the specific time comes around again.
If there is no time zone set in the Contact Record, ONTRAPORT will use 10 AM in the account's time zone. If that is not possible, then 10 AM Pacific time will be used (but this is rare; we almost always have the account's time zone available).
To have a Wait hold the Contact until the NEXT time a specific time comes around, for example 5 AM, set the Wait for zero days, hours and minutes, and set the specific time. Contacts landing on the Wait from 12:01 AM to 11:59 PM on Monday will all get the email at 5 AM on Tuesday in this example.