Lidos are enjoying a surge in popularity in Britain at the moment (for instance, see this article). There was a piece about them on a radio programme a few days ago, and I noticed that roughly half the number of people who spoke pronounced the i of lido as ee, and the other half pronounced the vowel sound like eye. The OED offers both pronunciations as correct.
The word lido comes originally from the name of a spit of land near Venice, the Lido di Venezia. Europe's first sea bathing resort was established there in the 19th century, and it became very fashionable. There are citations in the OED from the 17th century, and these refer to that region. It took on a wider meaning in the 20th century. A citation from a 1930 newspaper that is in the OED has lido in inverted commas and with a capital L - "The question of the safety of bathers in the Serpentine ‘Lido’ was raised at an inquest..yesterday."
The Italian word lido is from the Latin for shore, litus.