Is the phone completely broken? Or if you let it dry for a few days, maybe it will start?
Contacts are not stored on simcard anymore, they are on the phone's flashdrive.
But since it's static, a bit of water should hurt it.
Go to iCloud.com and login with the credentials your son used for iCloud on his 5C. Then go to Contacts and see if his contacts appear in there.
If so, simply login to iCloud on the 5S with his credentials and all his contacts should appear.
If above is not the case, then I'm afraid there is no way to get the contacts off of the 5c.