Black is a great idea. I'd say nothing too logo-heavy or laden with big, flashy hardware or fringe. If you don't need to carry papers I'd also suggest a hand bag rather than a shoulder bag. Sometimes shoulder bags can throw off the drape of your suit jacket. If you carry files around (in the future) then all bets are off and a black tote is your most practical choice.
You might see logo bags carried by the female associates - it depends on your firm's culture - but that doesn't mean that you should do so. I'd say lay low the first week and get a feel for your workplace. Sometimes a young intern who has flashy bags and jewelry sends the wrong message, kwim? Even if the associates like your look the other interns might resent you, and this could really hurt you.
Some more expensive black bags that can give you an idea of office appropriate are the Mulberry Bayswater (nice because it can be shoulder or handheld), LV black epi leather (because it's so low-key, but high quality), and some J.Crew satchels like the Brompton hobo (but only if it looks proportional on you - if it looks like a suitcase I'd pass), and Tory Burch's Robinson line in black.