I live in NY and will soon have to face this myself. There is no way around publishing your name in a newspaper because it is required by the state; however, it doesn't specify which newspaper. Therefore, if she could find a small paper that has a wide enough circulation to meet the judges criteria, she might be able to avoid it being posted online. Most of the bigger newspapers in NY have digital copies, but it isn't definite that the smaller ones will post everything online. Also, it would be unlikely that anyone would pick up on if it were in a small paper that few people read. Whatever she does, don't let her choose something like the Time or the post. Those are read state wide and have digital copies.
By the way, all name changes in NY are part of the public record (unless you are changing your name for safety reasons). So, no matter what she does to cover her tracks, there will always be a record of the change. It sucks that the state couldn't be more sympathetic to trans people, but there isn't much we can do.