Can you use a mana ability to pay for the ward cost when a spell targeting your permanent is cast?

Ward is a triggered ability that triggers whenever the permanent with ward becomes the target of a spell or ability an opponent controls (CR 702.143a). When that trigger resolves, the opponent must pay the ward cost or the spell or ability is countered.

Mana abilities are special: they do not go on the stack and can be activated any time you have priority or are asked to pay a cost (CR 605.3b). When the ward trigger resolves and the opponent is required to pay the ward cost, they may use mana abilities at that moment to generate the mana needed, just as they would when paying any other cost during resolution.

It is worth noting that ward's trigger goes on the stack after the spell that caused it (CR 702.143b). Both players receive priority in normal stack order. When the ward trigger eventually resolves, the game asks the controller of the targeting spell to pay the cost; they can tap lands or use other mana abilities right then to produce the required mana.

Concrete example: Your opponent casts Lightning Bolt targeting your Grizzled Huntmaster, which has Ward {2}. Ward triggers and goes on the stack on top of Bolt. When the ward trigger resolves, your opponent can tap two lands to pay {2}, satisfying the ward cost and keeping Lightning Bolt on the stack. If they cannot or choose not to pay, Lightning Bolt is countered by the ward ability.

So yes, mana abilities are always available when a cost must be paid, including ward costs (CR 605.3b, 702.143).