There is now an ever increasing number of restaurants and bars on booming Hydra. It´s probably best to discover yourselves, though the ones mentioned in the posts above are good hints.
To us the main distinction for food quality was, is the fish and meat fresh (good) or frozen (bad and dry). Most restaurants on the island are reasonably priced with mains from 8 to 15 €, fresh fish by the kilo a bit more. Even the few more upscale ones have the mains mostly under 25 €.
I try to list here the restaurants and bars to my knowledge related to Leonard Cohen. Mind, obviously they all have changed quite a lot since Cohen´s times. And please correct me where I am wrong or add more information.
Duskos on a romantic village place inside the town, the one with the tree, you all know the foto Cohen playing guitarre there, still good food at reasonable prices
Cristina & Manoli's, now To Gitonikon near Duskos, mainly on a roof terrace, insiders know at which table Cohen sat, after the horrible crime now seemingly run by the sons, like Duskos still good food at reasonable prices
Roloi at the port under the clock tower, our home base of Nikolas, Nota and Iannis, the former Katsikas shop and bar, where Cohen first met Marianne as it is said, at the Roloi one can sit from breakfast till very late and finds also a menue to eat, the hospitality of the owners is exceptional
Bill´s Bar is now integrated in the Bratsera hotel also close to Duskos, the bar is open also to non hotel guest, you could have dinner by the pool in the courtyard or rather just have drink inside at the original bar
Pirate Bar in the corner of the port, close to the Roloi is said to have been another hangout of Cohen and friends
the red and yellow house at Kamini port is told to have been another bar that Cohen frequented, but it´s closed since years
1979 Kassel / 1980 Munich / 2008 Munich, Hamburg / 2009 Cologne, Antwerp, Lisbon, Budapest, Barcelona / 2010 Salzburg, Florence, Marseille, Stuttgart, Bratislava / 2012 Ghent, Berlin, Verona, Barcelona, Madrid, Lisbon / 2013 Paris, Mannheim, Rome, Prague, Pula, Zuerich, Amsterdam
2011 Haiger / 2014 Dublin / 2015 Hydra, Viladrau / 2016 Amsterdam / 2017 Hydra, Balaton