I prefer
Elowen. I don't really think of LOTR; I think the L makes it sound a lot different from
Eowyn.
I like names like
Eowyn and
Arwen because of LOTR but also for the pseudo-Celtic vibe they seem to have, which is why I like
Elowen.
Elowen Kate and
Elowen Violet are pretty. I wouldn't use a nickname for it but
Elle,
Ellie,
Ella,
Winnie,
Wyn (for Elowyn),
Wen,
Wendy, etc would all work.