Is there anything the anesthesiologist can do to reduce how sore my 16 year old gets from being put under?

Yes. If you mean muscle soreness yes. ask the anesthesiologist not to use the drug "succinylcholine"
Good pain relief. What type of surgery is planned? Depending on the answer, different types of pain relief may be in order. Sometimes, a muscle relaxant called succinylcholine may cause muscle soreness; this medication can often be avoided as we have other choices available. Unfortunately, most types of surgery usually cause some postoperative pain. Talk to your anesthesiologist to plan the best pain relief plan.
Sore where? If it is an overall achy muscle feeling, then he may have had succinylcholine, a muscle relaxant. There are alternatives, but they might not be appropriate for him or the surgery. Also can get sore being stuck in same position for a long time. Unfortunately not much can be done about that, even with good padding. As for surgical pain, options depend on the surgery and patient.
Talk to MD. Some of the medications we give to patients can cause muscle soreness, as can the surgery itself. Talk to your child's anesthesiologist and see if there are specific things that can be done to prevent this from happening.