Either 1" or 11/4" will work. If you are using a pneumatic stapler, it will sink the staples into the top layer a bit, anyway, so the staples will still grab plenty of material even at 1".
What is the proper depth to which staples should be allowed to go into the material beneath the new underlayment? Or, what size staples should I use? The original floor is 3/4", the original underlayment is 1/4", the original vinyl floor is about 1/8" or less, and the new underlayment going on top of the old vinyl is 1/4"... resulting in a total layer thickness of about 1 3/8". Do I use 1 1/4" to penetrate almost all the way through the original flooring, or will 1" suffice to have the staple stop halfway into the original flooring? I appreciate the advice of professionals! Thanks!