Note that, if you're recording a sine wave of known distortion (ideally, none -- you can use a high order, low distortion lowpass or bandpass filter to assist in this), noise can be integrated out because it is incoherent and averages towards zero, while the distortion (which is phase locked to the fundamental) remains coherent.
You can also try testing it with a very linear ramp, which can be generated with a very stable square wave and an integrator (precision op-amp with high quality feedback capacitance -- polypropylene, polystyrene, teflon, or better still). Obviously, a least-squares fit to the resulting data series easily finds errors (INL/DNL), and the line equation can be compared against the input to find gain and offset error (but now you need calibration).
The frequency used for the ramp will still matter, because inevitably there will be dielectric absorption and loss in the capacitor, which will skew the slope ever so slightly. It should be as fast as possible (to approximate absorption as nil), but also as slow as necessary (ideally, hitting every code on the way up and down, thus proving a one-to-one transfer function with no missing codes -- assuming your design is intended to achieve this result).
Data arbitrarily close to the peaks (direction changes) should be discarded, due to finite rise/fall time of the square wave, ringing and settling, and finite frequency response of the op-amp and signal path. (Ideally, a triangle wave has harmonics with amplitudes of 1/N^2, for N up to infinity; a real triangle wave will have some harmonics not exactly that amplitude, but higher or lower than others, and also phase shifted; and after some N, the amplitudes will start dropping much faster than 1/N^2, perhaps N^3 or N^4, before becoming nonexistent for all intents and purposes.)
Tim