I've coached youth track, cross country, and running club for an elementary school. The serious replies so far are on the mark. Just listen and watch for signs of being tired or wanting to stop. Keep it fun and no pressure, and he should be fine.
More than distance, I would think about frequency and wearing decent shoes. Recovery is key, and for the youngest runners, we never went more than 2-3 days per week. That gives plenty of time for recovery and keeps it fun instead of becoming a chore. Then it's a matter of following all of the well-known rules about starting with shorter distances and frequent breaks, eventually building up to longer times with shorter breaks.
It's repetitive stress/strain that we have to keep an eye on.
Kids are stronger and more resilient than we give them credit for, and a miles is not really that far. We had kindergartners able to run a mile with no problem and most could run 1.5 miles.
Make sure he's wearing running shoes or something with at least a little bit of structure (i.e no crocs or sandals) with laces and not velcro straps. This is mostly to keep the shoe on the foot (and prevent injury from falling or twisting an ankle) as opposed to some philosophical idea of minimal/maximal footwear for running.