I think it really depends on the turbo... for IHI turbos, just the model number is enough. For MHI you need at least the compressor (16G, 18G, 20G, etc) and the hotside (TD05, TD06, TD06H, etc). For Garrett it gets a little more complicated, but knowing the GT28/GT30/GT35 plus the compressor capacity in lbs/min plus the turbine AR (.63, .82, etc) would do a reasonably good job.
Garrett part numbers would be useful, but the commonly used designations like GT30 are meaningless and the meaningful part numbers aren't commonly used. Look at all the numbers at this page and you'll see what I mean: http://www.turbobygarrett.com/turbob..._700382_12.htm