Three are many different possible explanations for what happened to your son, but there is absolutely no way any of us here can tell you what it is. I am very sensitive to dental work and often end up with swollen lips and even bruises after certain dental procedures, but me sharing this does not help alleviate your son's pain or help you figure out what's going on. I say that not to be dismissive or disrespectful of your question, but only to highlight the fact that we can all share our stories here, but not one of them will help you and your son's doctors/dentist figure out what happened.
He needs to be seen by the pediatrician or dentist for his condition to be properly diagnosed. This is important so that you can take proper steps to diminish the possibility of this happening in the future and to know what to do to alleviate the pain if it does happen again when he has dental work.
Kids often do bite their lips when they are administered novocaine because they can't feel when they're biting. It's a very common thing, even when parents are on the lookout for it post-procedure, and that's probably why the dentist mentioned it. I am NOT saying that this is what happened with your son, but it does happen quite commonly and thus would logically be considered as a possibility. Again, the only way to know is to have your son evaluated in person.
There are many different types of blisters that can occur around the mouth area. Herpes simples viruses (which cause cold sores to erupt) are just one of the possibilities, but they are one of the most common.
The vast majority of the American population has been exposed at some point to the various herpes simplex viruses. After exposure, the virus lives dormant on the nerve endings until something triggers an outbreak. We have lots and lots of nerve endings around our lips and mouth, so this is one of the places where the sores show up. It is also the reason that the outbreaks are so painful . Some people who are prone to cold sores find they have outbreaks after dental procedures. However, whether or not your son's condition is an actual outbreak of cold sores should be left to your doctor/dentist to diagnose.
If it is a herpes infection, you want to know so that you can take steps to minimize outbreaks during future dental procedures. Herpes infections are very painful and very contagious. It's important to know how to manage these outbreaks so the infection isn't further spread to other sites on the mouth and face or spread to other people. There are medications that can be prescribed to minimize outbreaks, but they must be monitored by a physician. Dentists can also prescribe these medications.
If your son likes popsicles or otter pops, you can give him those. The coolness of the pops may alleviate some of the pain temporarily. It's good that you're taking him in to be seen. Once you find out what's really going on, you'll feel much better and will have a plan for how best to help your son.