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Just Hazel for me. Hazell just looks awkward because that's not how the word is spelled. Hazelle kind of looks like a typo for gazelle (especially since H and G are together on the keyboard...) Also, Hazell/Hazelle/Hazele would be haz-ELL, not HAY-zel.

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Haze and Hazey are cute. So are Elle and Ellie, but more of a stretch. Zel, Hay? (Okay, Hay wouldn't work because she wouldn't know whether you were saying "hey" or her name...)
I think Haze and Hazey are the best nn options here.