Brown is a pretty unusual color for a living plant, though it's such a common color for dead ones that one wonders why more living plants don't try it, for the camouflage.
This is a particularly tough category, so some of these are a stretch, and others aren't a stretch but the pictures don't show it well. If anybody has some recommendations for brown-leaved plants that I've left out, I'd be happy to hear them.
Cryptanthus cvv. (some cvv.)
Kalanchoe tomentosa 'Chocolate Soldier.'
Philodendron 'Prince of Orange.'
Pilea 'Moon Valley.'
Solenostemon scutellarioides 'Tilt a Whirl.'
I have varying levels of experience with the above set of plants, so my recommendations may be less reliable than usual, just so you know. The three of the above I would recommend most strongly would be Alworthia 'Black Gem,' Philodendron 'Prince of Orange,' and Vriesea splendens. None of them are particularly good in low-light situations, but that's true of everything on the list.
My favorite is the Alworthia: I've had one (and then, later, several) for a long time, and I can't recall them ever being any trouble. The brown color doesn't develop without pretty intense light, but even when they're just green, they're good plants.
Second favorite would have to be Vriesea splendens, which is maybe not the brownest of the plants on the list, but it's also pretty easy to grow. Easier than its reputation would have you believe, anyway.
Third would be Philodendron 'Prince of Orange,' which is not, technically, a plant I've grown, but I've had the related plants 'Moonlight' and 'Autumn,' and those have both (eventually) become well-behaved citizens of the plant collection. Light was apparently the main issue, though temperature might have been a problem also. 'Autumn' was all but brought back from the dead, i.e., it was restarted from cuttings, and 'Moonlight' has always been pretty-well behaved, but it turned into a rock star when moved to the plant room about a year ago. (Fertilizer, it turns out, is also helpful. I should do a fertilizer post sometime.) I have no reason to think 'Prince of Orange' is significantly worse than the others. The brown coloration is very temporary, and happens somewhere during the leaves' transitions from orange (brand new) to green (old), but if you're not too picky about the definition of "brown," it's in there.
The anti-recommend probably ought to be Lithops spp., but I've been hard on Lithops lately and feel bad going after it a second time. So instead, I will un-recommend Solenostemon scutellarioides 'Tilt-a-Whirl.' I managed to keep a number of coleus indoors over the winter, even propagating them as I went, but even with lots of artificial light, I was unable to keep them from getting leggy and hideous on me (particularly 'Splish Splash'). Though they can be overwintered indoors, and extremely motivated persons can grow them under lights year-round, I don't recommend bringing them inside if there's any chance of growing them outside.
Calathea makoyana, a little
Codiaeum variegatum, some cvv.
some Episcia cvv.
some Gasteria cvv.
some Kohleria cvv. (thanks, Andrew!)
some Pelargonium x hortorum cvv.
Philodendron 'Autumn' (in the same way as for 'Prince of Orange')
Pilea involucrata 'Norfolk' (?)
Polyscias scutellaria, a little bit
Sansevieria kirkii var. pulchra 'Coppertone' (thanks, Karen!)
Zamia furfuracea (in high light)
Zingiber malaysianum (under certain viewing conditions)