This is more common than most families realize, and in many cases, it has nothing to do with anything your son did wrong.
Federal intake at a new facility frequently begins with a short stay in the SHU, and the reasons given can feel frustratingly vague. Missing paperwork, a bed shortage in the designated housing unit, a custody level adjustment that has not been fully processed yet, these are all legitimate administrative reasons that result in a new arrival sitting in restrictive housing while the facility catches up on the backend. It is bureaucratic rather than punitive in most cases.
I went through exactly this at a federal camp. My designation was to the camp, the lowest security level available, and I still spent two weeks in the SHU when I first arrived simply because they were not ready to move me to where I was supposed to be. There was no infraction, no investigation, no disciplinary reason. The system just needed time to process everything and that was where I waited. It was not fun, and the uncertainty of not knowing how long it would last made it harder than it needed to be.
The timeline varies but a week to two weeks is common for administrative placements. If it stretches significantly beyond that, it is worth having him ask his case manager directly what the specific hold is and what needs to happen for him to be released into general population. He has a right to know the basis for the placement and an approximate timeline.
Keep the letters going during this stretch. Mail reaches the SHU even when everything else is restricted, and knowing someone on the outside is paying attention matters more than usual when you are sitting in isolation for reasons that were never fully explained to you.