We at Kythera-Family.net are putting together a map of all the fresh-water sources on the island. You can view the current map we have here:
www.kythera-family.net/download/WaterSprings1.jpg

So far we have the following sites:
1. Kaloyerou behind Agios Nikolas
2. Petrouni
3. Agios Mama (half an hour from Petrouni)
4. Routsoinas (apo to Kalakathi)
5. Keramari in Karavas
6. Amir Ali
7. Portakalia
8. Milopotamos/Neraitha
9. many pools below Neraitha
10. "Galani" on the road down to Pelagia from Potamos.
11. Brisi at Mitata
12. Mudaro at Mitata
13. Gonia between Paliopoly and Mitata
14. Brisi at Viarathika
15. "Sithero-Nero" near Diakopoulanika
16. Kavpi behind Diakopoulanika

Please help us find more. Just send a mail to james@kythera-family.net to contribute!

Suggestions:

Eleni Haros:

ΒΡΥΣΕΣ ΚΑΙ ΠΗΓΕΣ ΣΤΑ ΚΥΘΗΡΑ
Στα Κύθηρα υπάρχουν περισσότερες από 70 πηγές. Οι περισσότερες λειτουργούν μέχρι σήμερα (από τις πιο γνωστές είναι του Καραβά, του Μυλοποτάμου, των Βιαραδίκων, των Μητάτων, της Αγίας Πελαγίας, της Γουρίας στα Λογοθετιάνικα, της Γωνίας, των Τριφυλλιανίκων) Κάποιες έχουν στερέψει (στα Κουφάρικα, στους Φούρνους, στο Καλαμίτσι, στο Καψάλι το νερό της Αφροδίτης) Άλλες είναι υποθαλάσσιες (στο Διακόφτι, στον Αυλαίμονα, στη Φελωτή) Κάποιες είναι σιδερόνερα με εξαιρετικές θεραπευτικές ιδιότητες (Πετρούνι, Άγιος Μάμας, των Οχέλων, των Μαγγάνων στον Καραβά) Η πηγή στου Λυκοδήμου είναι μέσα σε σπηλιά.

Translation:

On Kythira more than 70 springs. Most are still flowing today (one of the most famous of Karavas, Mylopotamos, Viaradika, Mitata, Agia Pelagía, Gourias in Logothetiánika, Gionia, Trifyllianika), some have dried up (in Kouphárika, at Fournos, the Kalamitsi, in Kapsáli "the water of Venus") others are underwater (at Diakofti, Avlemonas and at Feloti Bay). Some are mineral springs with exceptional healing properties (Petroúni, Agios Mamas, Ochélōn, Mangánōn of Karavas), and the spring at Lykodemou is inside a cave.  

From Tony Zaglas

I think there is one at Tryfilianika (never been)

There was one at one at Vrohandarika (on road between Tryfilianika and Paliohora)

There is one at Stavli in the langathi north of Agia Marina.

There was a spring at Skotini NE of Potamos (Kastelous)

There was one at Petroulou in langathi east of Agia Anastasia (blocked up)

There was one in the langathi of Agia Pelagia (Tsagourna) there was some sort of water conduit possibly for an old  water mill.

There was one at Kalamitsi on the road between Agia Pelagia and Kaky Langatha

There should be springs below Mertidia, Kosounari, Logothetianika and Dourianika. Check where the streams are in the map on KFN. A few may be seasonal.

From John Vanges:

There also is a very old spring down on the right of Bayianika where the gardens are, also called 

Sithero-Nero. In 1965, if you let a glass of water from there stand for about 10 mins. it would 

form a red slush deposit at the bottom of the glass, as demonstrated by my papou, Bagoyianni.

From Matina Vrontas

As I recall, there was a fresh water spring pouring into the sea at Nisakia which is on the coast, west of Logothetianika.

I walked there 40 years ago from Dourianika but have not been back since. 

Clearly not easily accessible but useful to know in terms of the listing.