First filter the analog input. Analog filtering will lead to creep so you must use a FFT and compare the high and low frequency components (how depend on many factors, you must read and record and then design and code). Second, have a ADC resolution 16 times your required weight resolution and whenever your reading is less than minimum (1/10 oz I think to recall) assume zero and adjust accordingly.
Who said it was easy, but if it was it would'nt be fun
Erik