Honestly OP - and other posters - this is another area of anti-Irish undertones that people don't even recognise.
Brian is a common Irish name. Its not as popular as it once was, but we have a culture where naming babies after adults is more common, and so I would be a bit surprised but not totally shocked to come across a baby Brian.
There is much more a tendency to have 'trendy' names in the UK, and unfortunately Brian as one that travelled must have fit that category and is linked with a particular generation. But - would you mock a baby David? Must have been similar to Brian in age and popularity. If not, do you think why?
In 2015, Brian was in the top 100 in Ireland. Its only recently gone outside. So all this people laughing hysterically - do they imagine they won't meet any Irish people in 20 years? There is so much small-minded parochialism, but I bet if someone said 'I'm Indian, I want to name my son after my Indian grandfather, people are mocking the name and my husbands family say they can't pronounce it' people would be rightly calling out the family for their racism and prejudice.
Your extended family need to wind their necks in, and you need to protect your baby. I don't think the responses you get here are reflective of real life. I know some insane names, and I would personally much prefer a baby Brian to a baby Wilbur, or Stanley, or Archie.