This appears to be a term for a surgery done while the patient is awake, either under local anesthetic or no anesthetic.
I couldn’t find a formal definition, but I found a bunch of usages where it seemed to fit:
From various sources:
Reportedly, following a further waking surgery during which Zanardi was alert the whole time, he was able to speak with his family.
[…] Waking surgery has been done in operating theaters for patients who are allergic to anesthesia.
[…] Various methods were tried to relieve the pain and anxiety of waking surgery, including mesmerism and other experimental techniques,
[…] 3D glasses can be worn by patients for waking surgery, or by surgeons to “model” the patient and gain in precision
I also found another usage to mean a situation where a surgical patient inadvertently wakes up during the procedure.
I asked a friend who is a nurse anesthetist, and they responded:
Hmmm. Not sure I totally get what this refers to. “Waking surgery” is not a term I’m familiar with. […] There are surgeries where we wake the patient in the middle or keep the patient awake (surgeries on the motor strip of the motor cortex, for example), but that anesthetic approach is rare.
So, I don’t think this is a formal term.
Why I Looked It Up
I found this in an article about former race car driver Alec Zanardi (the first quote above). While I was searching, that same article was in many of the search results, which makes me feel like this is not a very commonly-used phrase.