The quality or state of being elegible; however, this word is extremely rare and may be a variant or error for eligibility (the quality of being qualified).
This appears to be a misspelling or variant of eligibility (from Latin eligibilis 'able to be chosen'). If intended as distinct, it might derive from elegy-related roots, but no standard dictionary recognizes elegibility as a standard English word.
This word is so rare it barely exists in English—it might be a ghost word that appeared in one old dictionary and never caught on, or possibly a variant spelling someone tried to make sound more like elegy-related words.
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