My son wanted to change his sink bowl and cabinet combination to a pedastal sink to take up less room. It looks like a 9" wide pedastal would cover up the plumbing but I'm not sure what the elbow and blind cap are on the top of the drain. This is an old house and I am sure this is the 100th time it has been renovated. Is this capped off section what should have been going to a vent? If it has no function, can it be eliminated?
That appears to be an Air Admittance Valve, (AAV) also known as a mechanical vent or Studor vent (the latter being a brand name.) i.e. not a cap.
It's a one way valve (if working correctly) that serves to vent the trap. If renovating in such a way that running a proper vent line is possible, I'd highly recommend doing that. Otherwise you need to preserve or replicate this in the new setup. Make sure that it can be accessed, since they often need to be replaced when they stop working correctly.
Since it wants to be located as high as possible (thus, correctly right up under the cabinet top in this "under sink, out of wall" install) one approach would be to put it in the wall cavity behind an access panel, between the medicine cabinet or mirror and the ceiling.