What does Aydan mean?
Aydan is a Turkish/Azerbaijani name from 'ay' ('moon') plus the suffix -dan ('from'), i.e. 'from the moon.' Some sources list unrelated secondary meanings ('tall palm tree'), but the moon reading is primary.
What does Esmeray mean?
From Turkish 'esmer' meaning 'dark, brunette' and 'ay' meaning 'moon', giving 'dark moon'. (A 'sunbeam/beloved' reading circulates but does not fit the Turkish elements and appears to be a folk reinterpretation.)