I admit, I fought with this same issue on my kids (another child of the 80s here!). I finally decided that since my kids are quite young (preschool age), they would be with me or my husband 24/7, so how could someone knowing their name matter? My kids are also the type to tell their names to everyone and anyone completely unprompted, so I don't think a shirt will exactly be letting the cat out of the bag.
While I don't have anything picked out yet, my kids will likely be wearing at least a couple shirts with their names on them during our trip. They LOVE seeing their names on stuff and having a cast member or character call them by name will make their day!
That being said, if my kids were old enough to be apart from us in public places, I think I would feel differently. While I hope by that time to have taught the kids to think critically about situations involving strangers, being lost, etc, I would definitely feel less anxious about them being apart from me without their names being plastered everywhere.
My thought is to follow your gut. If it's going to make you anxious the whole trip, don't. If they're going to be with you or another responsible adult the whole time and you're cool with that, go for it - it's a neat memento.
Now, if only I could figure out what I'm supposed to do with their Halloween candy! The after school specials made it clear that adults should check it, but how exactly do I do that?!

Eating the candy myself seems best, but for some reason they didn't like that
