Second to the Cave of Drogarati nearby to Mellisani. But be prepared on the way of returning on the surface the route is tiresome. The kids would most definetely need to make a couple of stops at least ( if you take them with you)

To the lighthouse of Argostoli search nearby for the spot named Katavothres where you will witness the sea disappearing under the land.
Nearby Drogarati and Mellisani is a small village of Karavomilos which is essentially near the sea lies one of the exits of the underground river that begins to Katavothres. The other exit is into Melissani Lake.

To the southeast of the island there is a place named Kounopetra which means the ever lasting moving Rock. It's a huge Rock adjacent to the shore that before the 2014 earthquakes use to solely make small moves right and left. The explanation is that it sits to unstable soil that probably resembles like moving sand. There is no fear though on standing on top of it as thousands have visited and standed on to it.

Lastly the best attraction is the mountain of AINOS the highest one in Kefallonia 1628 meter tall. You will definetely need to hire a 4×4. In one of it slopes near a monastery there are wild horses as a heard. It's quite unique spectacle and very beautiful animals. Before visiting it though (if you decide to go) you should equip yourselfs with good jackets as the conditions change drastically over time.