I ran across this term yet again, in a New Jersey case decided just last week. Just in case some of you find it handy in the future — to understand, not to use — it means “charitable,” as in “a [charitable/eleemosynary] institution” or “an organization’s [charitable/eleemosynary] mission.” It’s more popular in legal circles, being (according to Westlaw) merely 30 times less common than “charitable” in court opinions these days, as opposed to (according to Google Ngrams) 150 times less common than “charitable” in books.
UPDATE: Commenter dw notes that it’s useful to have a pronunciation guide; here’s what dictionary.com offers, with the accent on the bold syllable:
el-uh–mos–uh-ner-ee
el-uh–moz–uh-ner-ee
el-ee-uh–mos–uh-ner-ee
el-ee-uh–moz–uh-ner-ee