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According to this website (it gives full videos of pronunciation and meanings):

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Meeli (f) is a newly-created name that comes from the Estonian word, meel, which means mind/thought/idea. It also says the name coincides with an Old Estonian name, but it doesn't go into further detail. I'm thinking the Old Estonian name being referred to in the video is Meelis (m).

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http://www.fredonia.edu/faculty/emeritus/EdwinLawson/estoniannames/index.html

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The meaning for Meelis is listed as "agreeable and pleasant" on this site: http://www.esis.ee/ist2000/einst/culture/people.htm

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I haven't been able to find anything else.