If I play Brynhir Thundersong, and my opponent reacts with Discipline, can he resolve Discipline to draw a card, and continue to react on the chain that Brynhir Thundersong started?

Yes, your opponent can resolve Discipline and continue to react to the chain.

According to the rules of chain resolution and the specific interactions with Brynhir Thundersong:

The Sequence

1. You play Brynhir Thundersong: She resolves immediately to the battlefield. Her "When you play me" triggered ability goes on the chain.
2. Opponent reacts with Discipline: Your opponent adds Discipline to the chain on top of Brynhir's triggered ability.
3. Chain Resolution: The chain resolves in LIFO (Last-In, First-Out) order.
   • Discipline resolves first: The unit gets +2 might, and your opponent draws a card.
   • Window of Opportunity: After Discipline resolves, there is a window to react before the next item on the chain (Brynhir's ability) resolves. If your opponent drew a reaction card from Discipline, they may play it now.
4. Final Resolution: Once all reactions are played and the chain is empty of other items, Brynhir's triggered ability resolves.

Important Nuance

• The "Shut-off" Timing: Your opponent can continue to react to the chain until Brynhir's triggered ability fully resolves. Once that ability resolves, the restriction "opponents can't play cards this turn" takes effect, and they will no longer be able to play any further cards from their hand for the remainder of the turn.

As noted in the official FAQ: "Players can play reactions during chain resolution, not just before it starts. After each chain item resolves, if more items remain on the chain, priority goes to the controller of the next chain item, and both players can add new reactions before the next item resolves."