You can register now and then change the dates on Monday with no fee. I registered for the first session and will probably take time off work for that (depending on my internship dates), but I’m local. Most NM kids I know are registering for the first orientation date, and most OOS students I’ve talked to are registering for the last.
As far as Honors priority registration goes, I believe that only is in effect starting with registration for spring. Per https://honors.unm.edu/academics/registration.html, peer advising is required for early registration, and since the earliest that can happen is at orientation, it wouldn’t help.
Registration is open through the 2nd week of classes, and professors can give overrides for a few weeks past that.
Unfortunately, none of my classes are freshman classes, and they will already be full by even May 30th. If he is registering for regular frosh classes, he should be fine.