Can you respond to your own triggered ability by activating a mana ability before it resolves?

Mana abilities are special: they do not use the stack and cannot be responded to (CR 605.3b). However, this cuts both ways — you can activate a mana ability essentially any time you have priority, including while a triggered ability is on the stack waiting to resolve.

When a triggered ability goes on the stack, players receive priority before it resolves (CR 116.3b). During your priority window you may activate any mana ability you control, because mana abilities can be activated whenever you could activate an instant — and even at additional times (CR 605.3a). Activating a mana ability doesn't itself use the stack, so it completes instantly without giving anyone a chance to respond.

This is most relevant when a triggered ability requires a mana payment upon resolution (like cumulative upkeep or an 'unless you pay' trigger). You don't need to have the mana ready before the trigger goes on the stack; you can tap lands for mana the moment you're about to let the trigger resolve.

Example: Your Braid of Fire triggers at the beginning of your upkeep, adding red mana. Separately, suppose you have a 'pay {2} or sacrifice a creature' trigger on the stack. Before you let it resolve, you can tap two lands for mana right then — no prior setup needed — because activating those mana abilities requires no special timing window beyond having priority.

One small caveat: mana produced this way goes into your mana pool and is available for costs paid during that trigger's resolution. Mana pools empty after each step and phase (CR 500.4), so the timing must be tight — but activating mana abilities just before a trigger resolves is perfectly legal and very common.