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I'd go with Jenna, because it's not so far off meaning as Jennifer is. In Jennifer a Jayne would have a different name, whereas historically many Janes have called themselves Jenny, and Jenna's not so far away from that.
Did you know that many Jewish girls called Sheine (Shayna) anglicised their names for general use by calling themselves Jane? I think turning Jayne into Shayna would be a bold move worth considering - the name is beautiful, means "beautiful," and seems suitable for general use these days. It would be a great way to honour a Jane/Jayne.