Thanks for all of the input everyone. I'm thinking that it probably was at one time used to feed something else like an icemaker. We do have a fridge with a water hook up and icemaker. We also have a dishwasher, and the undersink plumbing does have a trap... but this wasn't for any of those. So I looked a little more closely, and you can see the a hole in the cabinet base with the same 1/4 inch blue tubing just below the surface. It has been cut and re-routed directly into the drain. Someone must have been using it for something else and done the "ah screw it" method, by just cutting the line and feeding it into the drain system, because the line litterally looped directly from the copper supply to the pvc drain system. If I would have opened tha valve on this line it would have been constantly flushing water down the drain.
So I removed the whole supply line at the shut-off and replaced it with braided stainless, and also changed out the pvc parts where the line had been tapped into the drain, effectively removing the whole contraption. thanks again!!!