My opinion: definitely Barron, by process of elimination. Brighton is great in theory, but it bothers me the way people will say it differently. Some will pronounce the T on the tips of their tongues, while others will swallow the T in the backs of their throats. Barrington is super cute (reminds me of Paddington Bear!), but looks too much like a surname to be a first name to me. Barron is perfect and would work great from infancy right through old age. Easy to pronounce and spell. Will never be confused with your surname (ie, people having to ask "wait, was his name Barrington Spencer or Spencer Barrington?").
Barron is a Trump name, and of course a hereditary title as cosmonaut pointed out-- like Earl or Lord.
Brighton is a seaside town on the south coast of England that's synonymous with cheap thrills, kind of. It's like naming your kid the equivalent of "spring break!" or "Vegas!"
Barrington, unless a family name, is frankly OTT as a given name. It's like a caricature of aspirational yuppie names.
I don't want to be critical-- here are some more realistic suggestions that I think have the vibe you're looking for: