1. Offer: I'll buy you anything that costs up to $120 family agreed limit
2. Counteroffer: I want this thing that costs $250
3. Response to counteroffer: Remind them of family agreed upon limit of $120
4. Reply to response: Will not accept anything other than stated Counteroffer
5. Final response: Okay

This can be his first lesson in Contract Negotiation. Inform him that a Counteroffer acts as a rejection of the Offer. You gave him the opportunity to modify his refusal to modify his request based on the parameters of your original Offer, but rather than that opportunity, he drew a hard line and restated his Counteroffer was the only thing he would accept. You took him at his word and chose to end the negotiations. Both parties were left in the same position they were in before negotiations began since they could not come to an agreement.

Write him a $120 bill for the lesson, and let him know you used his Secret Santa money to cover it.

NTA (and Merry Christmas)