I did a lot of research on this a couple of years ago, and here is how I understand it (which doesn't mean it's correct

)
Traditional Spanish names contain two last names: the first is the paternal surname and the second is the maternal surname. Generally,
if a person is ok with dropping one of the surnames, it will be the maternal (second) surname that is dropped. However, this is not a hard and fast rule as some families choose to do different things in the name of assimilation.
I would say if you're labeling things before school, it's safer to use both surnames. If you
have to drop one, drop the final one. If you're teaching students to write their own names, ask them if they use both surnames.