Excuse me whilst I interrupt my usual Sophie and Dave snark to white knight.
Kids are cruel.
If they want to bully someone they will find a way. I was bullied for my name... My appearance... My level of intelligence... What I wore... Where I lived... My parent's jobs... I never fought back, but inside I would tell myself things that I knew where true 'I'm smarter than you' , 'I'll have a better job than you someday' and even petty things like, 'I have a nicer house than you'.
Sophie and Dave should concentrate on raising a little boy who is a good person with a strong sense of self worth. They can let the parents of future bullies worry about how they will raise their children to not pick on someone simply because of their name.
If they had chosen a name like Banana Brown, then maybe I would be concerned but Barney is fine, and I imagine it will probably grow in popularity as names do go in and out of fashion. 10 years ago Arthur scrapped into the Top 100 and I'm sure people who chose it for their son had people turn their noses up and say it's an 'old man's name', but now it's in the Top 10.