Greek Meaning: The name Eleni is a Greek baby name. In Greek the meaning of the name Eleni is: Light. Russian Meaning: The name Eleni is a Russian baby name.In Russian the meaning of the name Eleni is: Shining light.he name Eleni is a Greek baby name. In Greek the meaning of the name Eleni is: Light. Russian Meaning: The name Eleni is a Russian baby name.