Although it is undoubtedly a Greek name, Elena is also the closest possible (Greek) girl's name to Alan, the name of her biological father. Even though Melina thought he had abandoned her, she still named their child after him.
When Elena meets Haldane for the first time, he compliments her English; she replies that she wanted to learn French at school but her mother insisted that she learn English. She probably insisted on this because Elena is half-English, albeit unknowingly. Even though Melina believed that Haldane had abandoned her, she must have wanted her daughter to have some sort of link to the English part of her heritage, and the best way to do that was to have her learn the language.
We never see Elena interact with Katerina, her grandmother. This may be explained by the fact that Katerina rarely leaves her own home and Elena doesn't have time to visit as she has a business (the taverna) to run and a child to look after. Additionally, it could be inferred that Katerina dislikes Elena due to the latter being a modern woman who owns a business, in defiance of her grandmother's Stay in the Kitchen outlook. On top of that, Katerina is a Racist Grandma who knows that Elena was fathered by a foreigner, meaning she dislikes her grand-daughter for a reason that said grand-daughter is totally unaware of.
