Personally I would have picked the first one because it makes most sense.
You should see the comma as a place where you rest in a sentence, so it would be; "David told me about his new hobby, ........ which he's enjoying".
The second one does not really work. "David told me about his new hobby, that he's enjoying". It would work better if the sentence was "David told me about the new hobby that he's enjoying".
Then there is the third one, which does sound good with a break, but it should be a full stop; "David told me about his new hobby. He's enjoying it." You can see this because the way the sentence is started is with 'He's', which is pointing at David and not at his hobby, which means it has a different subject and thus is a different sentence.
Last but not least, number 4. This one just simply does not make sense in this context. "...his new hobby he's enjoying". To let that make sense it would be "David told me about the new hobby he's enjoying".
It is rather difficult to explain, because I do this mostly on feeling.