Can you activate a creature's tap ability in response to a spell that targets it before the spell resolves?

When a spell is cast and placed on the stack, both players receive priority before it resolves. This means you can activate a tap ability in response, as long as you meet the activation cost requirements at that moment (CR 116.2b, 602.2).

To activate a tap ability (written as {T}: ...), the creature must be untapped at the time of activation. The ability then goes on the stack above the spell, resolves first, and then the original spell resolves (CR 112.3, 405.5).

There is one important restriction: if the creature has summoning sickness — meaning it came under your control this turn — you cannot activate its tap ability unless it has haste. This rule applies to activated abilities that include {T} or {Q} in their cost (CR 302.6).

Concrete example: Your opponent casts Lightning Bolt targeting your Llanowar Elves. Before it resolves, you activate the Elves' {T}: Add {G} ability. The mana ability resolves immediately (mana abilities don't use the stack per CR 605.3b), you get the green mana, and then Lightning Bolt resolves and destroys the Elves. You successfully squeezed out the mana.

Note: Mana abilities resolve immediately without using the stack (CR 605.3b), but other tap abilities do use the stack and follow normal priority rules (CR 602.2).