Green’s Dictionary of Slang

brown v.2

also brown up, brownie up
[abbr. brown nose n.]

to act the toady, to be a sycophant.

[US]Baker et al. CUSS 89: Brown (up) [...] Brownie (up) Curry favor with a professor.
[US]H. Rawson Dict. of Invective (1991) 61: The sycophant may also be said to brown [someone], to get nose with [someone], to be a brownie (also with allusion to a junior Girl Scout), or to receive brownie points.
[US]Alt. Eng. Dict. [Internet] brown (verb) [...] to butter someone up in order to win a favor.