I suspect your BFs making fun of her speech pattern is just a consequence of him not liking her character. If he liked her, I doubt he'd say "well, I love the character, but hate how she speaks". When you dislike a character, every little thing about them becomes annoying.
I personally really like Bones, Brennan included, even if she does have her derp-moments. Like the death metal one where she tries correcting everyone's pronounciation of "skalle", with hers being the worst one.
I think my fave is from the same episode as the screen you posted, when someone recognizes her from TV
and asks "Didn't I see you on TV this morning?", to which Brennan, without missing a beat, asks: "How on earth should I know what you saw on TV this morning?"
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I'm bored out of my skull, let's play a different game. Let's pay a visit down below and cast the world in flame.