"Good kind of pain" is not a precisely defined medical term, but the concept is certainly a familiar one on several levels.
Pain is an incredibly complex phenomenon that indeeds needs dissection for analysis.
Depending on definitions there are a few kinds of "good pain" indeed. If you eat capsaicin your pain receptors get stimulated and after getting used to it you even start to enjoy the right dose of pain (even if it is in reality only your appreciation for your bodies response to those fake pains).
A similar response can be observed in cutting as a symptom of mental disorder or conscious body modification, or in certain techniques found in BDSM aficionados.
In all those cases one might argue that pain perception at the neurological level as such is really irrelevant and the psychological interpretation dominates the experience, which is then obviously found "good" by the participant.
Apart from philosophical considerations of what is "good" or "bad" you might also look at it from a biological, evolutionary or physiological functions perspective. What is useful is "good". How can pain be useful?
It keeps people from doing dangerous things. Alas, not from doing stupid things. But in sports there is for example the adage "no pain, no gain". This is often executed in a detrimental manner but hints at a connection to observe:
How to Tell the Difference Between Good and Bad Pain
What is good pain?
One of the most common forms of “good pain” is what doctors and physiotherapists may refer to as delayed onset muscle soreness,” or DOMS. It happens when you’ve challenged a muscle with something it’s not used to (new, returning or increased exercise). Within one to two days, you’ll start to feel soreness in the area and it may be tender to touch. But, it goes away quickly after that.
The pain comes from micro trauma in the muscle caused by rigorous exercise. But that’s not a bad thing. A muscle gets stronger, building denser tissue, when it has a reason to remodel itself. When it senses the tiny trauma, the muscle repairs tissue to allow for more endurance. The key here is the “micro” part of “trauma.”
(You need to compare that with 'Good Pain' Versus 'Bad Pain' for Athletes)
And most importantly, our conscious minds are often quite bad of arriving at the right conclusion through reason alone. The more basic snsation of 'pain' can be quite a good signal telling us what to do: nothing. Keep at rest when you are ill.
The anesthesiologist explained that during surgery and recovery I would be given strong painkillers, but once I got home the pain would not require narcotics. To paraphrase him, he said: “Pain is a part of life. We cannot eliminate it nor do we want to. The pain will guide you. You will know when to rest more; you will know when you are healing. If I give you Vicodin, you will no longer feel the pain, yes, but you will no longer know what your body is telling you. You might overexert yourself because you are no longer feeling the pain signals. All you need is rest. And please be careful with ibuprofen. It’s not good for your kidneys. Only take it if you must. Your body will heal itself with rest.”
The episode of that series might therefore refer to pain associated with diagnosed illness, associated with healing, not that strong and about to end soon.