My advice, show him change is as much an opportunity as it can be losing something and that change should always be met with some optimism. Then let him make his own decision. 

ok this is my view. Id do both. He needs to learn to teach thats what some of the 3rd dan is about.philosophy.

(you can out grow your master) lord knows i did.  

3rd dan is high but you need to keep good with his current master. thats half of climbing in rank.

(kukkiwan 3rd dan or not)? if he kuk stay close to that master

i would defanthly start his cross traing. there may be a teacher close that will teach more weapons and stuff.  . you can attend weapon clinics. at local schools. give that kid a weapon and let him start learning by his self. i learned most of my stuff on my own. latter was corrected by masters. they help me faster because they had something to work with.

but find out if his teacher can give him more classes if he knows hapkido gum do etc..there is no way i could teach all i know in my class but after yelp lots of stuff i can teach. to tell the truth tkd is easy to teach compared to kung fu. so i can teach tae kwon do all day but i get pissed teaching kung fu after a few hrs.

hapkido is awesome but you need to talk to his master let him know after his test and award ceremony after...after....after.  then let the master know he may know some one thats a friend. 

friendly things id take my son if i did not know the following is- boxing...muay thai.  the stuff you learn in there is awesome. boxing is a must for any black belt.

would also start competing if has not already done a thing. jr Olympics is cool and you get to travel.   plus you meet up with people all over the us. that has your skill and you feed and learn from them. awesome!!! of he does wtf do itf. two different worlds. with both ways he will be beyond.

hapkido is no joke. lots of falls lots rolls lots of pain.  know your son going to another school my mean his reduction in rank at that said new school. he may be use to his school and find it hard to git into a new grove. 

3rd dan is a rank you need to look back at your self. start finding what you cant and can do.  there are only so many kicks and so many one steeps. so many hand techniques. you have to hang in there with you master.  3rd dan is not a two time a week thing more like a once a week thing a the dojang. 

let him explore his self. with out a master looking over his shoulder. see what he is good at. 

Yes, learning decision making. 

Lots of good advice here.

I thinks it's important that he not become bored. Anyone can get bored by the time they reach 3rd dan but we all know kids struggle with it more than adults. If he feels that he's always learning something new, he'll stay enthusiastic. 

The fact that you could continue to work on the fundamentals and feel that there is still more to learn is part of the reason why a child cannot be a "black belt" until they turn 16.

There are a lot of life experiences we have to rack up to gain the patience and insight to be able to understand that even the basics can always be improved.

But even all that wonderful age and insight doesn't stop us from getting bored eventually too.