This doesn’t click in my brain

Your friend is correct. Silver Shroud Costume has an ETB ability that goes on the stack. As long as it hasn't resolved, it hasn't granted shroud to your creature yet (and it hasn't attached yet). While it's on the stack, your friend may activate Drowner of Hope's ability. It doesn't matter that there's already one instance of the ability on the stack; he may activate it again, and a new instance will be put on the stack. Do note that activating it again requires paying its cost a second time, so he has to sacrifice another Eldrazi Scion. (The first one was already sacrificed for the first activation, so he needs another.)

If he does so, the stack looks like this: (bottom) Drowner > Costume > Drowner (top)

Assuming everything resolves without any more response:

• The top ability, the second Drowner's, resolves, tapping your creature.
• The next ability, Costume's, resolves. Your creature gains shroud until end of turn, but it's already tapped.
• The bottom ability, the first Drowner's, fails to resolve; this is because your creature has shroud, so it's an illegal target. But it's okay for your friend; he already managed to tap your creature earlier.

EDIT: Rephrased and clarified things