Does Ward trigger if a spell that targets the permanent with ward was cast for free?

Ward triggers any time a spell or ability that targets the permanent with ward is cast or activated by an opponent. The trigger condition is simply that the permanent became the target — it does not care how the targeting spell was paid for. This is established by CR 702.167a, which states that ward triggers whenever the permanent becomes the target of a spell or ability an opponent controls.

Once ward triggers, the opponent must pay the ward cost or the targeting spell or ability is countered. The original casting cost of the spell — whether it was paid in full, reduced to zero, or bypassed entirely by an effect like Omniscience — is completely irrelevant to whether ward triggers and whether its cost must be paid.

This matters in practice: if a player uses Omniscience to cast Doom Blade for free and targets your creature with Ward {2}, ward still triggers. Your opponent must pay {2} or the Doom Blade is countered by the ward ability. The free casting does not exempt the spell from ward's requirement.

Note that ward only triggers for spells and abilities controlled by opponents (CR 702.167a). If you somehow target your own ward permanent with your own spell, ward will not trigger at all — but that scenario is unrelated to whether a spell was free.