Diana

I like this spelling over Dayana, because the "i" looks like a sparkle of a Diamond. Diana almost sounds like the precious stone in English and French.
The semetic meanings of Dayana, which has the same pronunciation as English Diana:
Hebrew from Dayan [ דיין --> דַּיָּנָה ]: Judge.
Arabic [ دايانا ] : root Dayan means "eternal" (well, the opposite of dying), also religions; faiths.
Apparently in Persian Dayana [ دايانا ] means "gold"
The Latin pronunciation Dee-anna does bring the meaning of ancient goddesses on the one hand, but on the other, from Dean (Latin): a church official, or valley (Anglo-Saxon)

Artemis, Apollo's twin sister. She was supposed to have helped deliver Apollo. She is the virgin goddess of the hunt.

Possible links to the Hebrew name Dinah. Dinah meaning "God has judged". Has a similar pronunciation with the modern pronunciation. Dinah is usually pronounced DEE-nah or DIE-nah while Diana is usually pronounced DIE-ann-a or DEE-ann-ah. They are very much the same and I believe that the old biblical hebrew and the old Roman merged to form the new modern name.

Diana also means "target" in Spanish.

Probably derived from an old Indo-European root meaning "heavenly, divine", related to dyeus (see ZEUS). --Taking this a step further, the Indo-European Roots section of my American Heritage Dictionary relates this to the root deiw-, to shine, from which derives sky, heavens, god. Their discussion of Diana regards the suffixed root *deiw-yo, 'luminous' to be the direct root of Diana and the variant *dyeu- the root of Zeus.