Unfortunately, the etymology geek in me will not be adequately satisfied. Wikipedia, font of all knowledge online, deems the origin of the name Braintree "obscure." Despite that, the online encyclopedia discusses several possible sources for the name, most of which support the idea that, somehow, Braintree began life as something like "Brantry" or "Branchetreu," both of which seem to mean "town by the river."
In fact, in the Domesday Book, a 1086 record of land use and taxation covering much of England, records Braintree as "Branchetreu." As far as I can tell, this appears to be the earliest record of the place name in its somewhat recognizable form. Thus it's worth looking into the sources of Branchetreu.
Branchetreu, like Braintree, breaks down into two syllables with a different origin for each: "Branche-" and "-treu." The speculation that Braintree means "town by the river" leads me to interpret the "Branche-" as equivalent to the French la branche, which is one of those words that means the exact same thing in both language. La branche in French and "branch" in English both refer to those small extensions of a tree growing up and out from the main trunk; both words also carry the same figurative meanings that denote the subsidiary parts of certain things [e.g., governments]. Therefore both words can mean "a separate smaller offshoot of a larger river." "Branche-" clearly equals "river," at least in my mind.
So what about that "-treu?" According to Wikipedia, the suffix "-treu" is equivalent to the modern suffix "-try" or "-tree," which used to mean "farm" and then expanded to mean "settlement" or "town." Apparently this appears in town names around Wales. If that's so, then "Branche-" = "River-" and "-treu" = "town," making "Branchetreu" = "Rivertown." The shifts changes in spelling and pronunciation we can attribute to the inevitable changes in language as it wends through the landscape of time.
Even though I know Braintree is basically Rivertown, the poetic images of its current iteration -- brains and trees -- will always teem in my mind. When I think of Braintree, I think of a tree in a cemetery growing out of someone's skull. More specifically, I think of an old New England family plot, full of effaced and canted stones, and an apple tree rooting in one corner, planted firmly in the pot of a dead person's skull. Or I think of another feral apple-like tree, once by a house that has long since disappeared. Short and broad, it bears the heavy burden of its fruit: bright ripe brains, swinging from their stems. Or, more metaphorically, I think of the nervous system as the epitome of a brain-tree: with the spinal cord as its trunk, it ramifies in electric branches throughout the body, with the brain at its fruiting crown.