[wilg]

It's not an incorrect word, but it can be possibly misconstrued as being infantilizing, because "child" somewhat implies a young person. But I can't think of a better word!

Perhaps it could have been more a bit more tactful to reword to avoid saying "child", like saying "your son is doing well" in this case.

[Symbiote]

Son or daughter avoids the possibility of infantalizing an adult child.

[wilg]

That’s what I said. Unfortunate that there is no obvious way to make it gender neutral.

[kbrannigan]

An adult child is still the kid of her parents.

[Symbiote]

Without further context, "his child was in an accident" suggests a child of ~1-12/16/18. (Younger than ~1 and we should say "baby".)

"His son was in an accident" could equally refer to an adult.