For a start, I'd give the gargoyle a breath weapon!
The thing that we normally associate with D&D gargoyles (a demonic statue on a rooftop) is actually called a "grotesque" in the real world; a fanciful statue. The word gargoyle comes from the French word for "throat", because the definition of a gargoyle is a decorative grotesque that is also a rain spout. All fantastical building statues are grotesques, but only those that pour water out of their mouths are actually gargoyles.
Hence, I'd suggest surprising the players with some kind of acidic breath weapon; perhaps even green slime. The gargoyle constantly oozes liquid from its mouth, and sprays either a stream or short-range cone of the goop at characters.
If your module supports a broader approach, then I'd consider introducing two types of gargoyle: the acid-spitting gargoyle (which, perhaps, cannot fly) and the non-spitting grotesque (which can fly). The two "sub-types" of gargoyle might hate each other with a passion. The gargoyles envy the grotesques for their wings, while the grotesques are infuriated that humans keep confusing them for the lowly gargoyles. Careful diplomacy might be able to play the two races off against each other, although given their embittered and evil alignments, the survivors are still likely to turn on the hapless adventurers.
One other possibility, again related to the fact that gargoyles are the rain-spouting variant: perhaps the party needs to talk to a gargoyle, as it holds a clue to a lost treasure or a mission objective. However, perhaps the party either doesn't speak Terran, or the gargoyle they need to speak to is mute. The players receive an obscure hint on how to communicate with the creature; "Look to its name". The trick is to restrain the beast somehow, and pour water through a vent in its neck. As the water emerges from its mouth, the trickling noise forms words in Common...