I am going to make a distinction between Kefalonian items i.e. indigenous to the island and shaped by its history and Greek items i.e. generally found all over Greece. There are details of where to buy most of these products within the website.

Local specialties :

Many different types of local herbs like oregano and mountain teas, checkout the shop on the island, their website
Olive oil: produced locally, from light to extra virgin and everything in between.
Honey: produced locally, flavored by the local flora, from mild to very strong flavors, but exquisite.
Wines: produced locally check-out the co-op.

Jewelry: exquisite gold pieces reproduced in Byzantine designs. Good News!! You can get a VAT refund at the airport, check out this site for details on how and what you will need.
Not all jewelers will supply you with the required paperwork, watch out for shops that display the ‘’Global Blue’’ sign. A jeweler shop that is very helpful is called “Kappatos” in Lithostarato. 

Sweets: Madoles & Madolato: a product of our Venetian past .
Icons of the local Saint, Agios Gerassimos, healer and miracle marker.
Lace, traditional embroidery, needlework and handmade textiles.

Street market:

When leaving Vivian villa turn right walk to the end the road and turn right again, you are now on the street commonly referred to as “the street with the palms”. After sunset part of this street is closed to traffic and many local artists and crafters set up stalls. There is often foodstuffs, paintings, jewelry, sandals, and bead work. 

General Greek Items:You can mix and match any wine in your gift back

Ecclesiastical paraphernalia.
Themed T shirts of ancient battles, Sparta is often popular with boys.
Reproductions of ancient leather sandals.
If you prefer a more substantial Ancient Greek sandal or high fashion shoes check out Migato in Lithostrato.
A converse type sneaker with a Greek flag motif.
Thongs with the a Greek flag motif.
Woolly slippers.
Shoes and seasonal fashion from Italy.
Greek coffee and coffee cup sets.
Copper coffee pots and grinders.
Long spout Olive oil pourers.
Ouzo, and Metaxa brandy. 

There are also many typical souvenirs shops on the water front, at the Agora, and in Lithostrato.
My shopping experience is clearly limited by my gender but let me assure you all the recommendations come on very good authority.