This always seems to be the big question - how to pronounce Brougham? We have a rich and varied language that has evolved over time with many influences and accents from around the world. We also know that in the 1800's there were many who could not read or write so many names were not accurately documented. Authors may also have put their own slant on names from their own experiences so names that were not Brougham may have been transcribed as such if they were similar. Different regional and national accents will inevitably have a big influence. Is it any wonder that we now have many ways to say a complex name like Brougham?
There are many pronunciations from around the World. In Brougham near Penrith it is pronounced Broom. In Huddersfield, 100 miles south east of Brougham, it is pronounced Brawm. These seems to be the two common pronunciations. In New York - BROCK-hummm is found, In Denver we have Brome. Early variants of the name suggest possibly other pronunciations - Burgham, Browham, Browgham, Brougham.
The first Lord Brougham emphasised that it was Broom, and early spellings of the name beginning with Brow- confirm this. Brow is pronounced Broo in the north of England.
I guess there isn't a right or wrong answer anymore. The important thing is that to be a true Brougham you must descend from Brougham village, south of Penrith in Cumbria, England. How you say that name is up to you - spelling though is a different matter!