Green’s Dictionary of Slang

to act the toady, to be a sycophant.

1967–8		Baker et al. CUSS 89: Brown (up) [...] Brownie (up) Curry favor with a professor.
1974		H.L. Foster Playin’ the Dozens 258: he suburban youngsters [. . . .] talked about nonphysical ways of ‘getting away with things’ [. . . .] Some of their suggestions included: 1. How to ‘brown’ the teacher.
1989		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.
1997–2002		Alt. Eng. Dict. 🌐 brown (verb) [...] to butter someone up in order to win a favor.