available day (for a venue or facility)
Refers to a day when a place (e.g., a meeting room, event hall, sports facility) is free and can be used.
Please check the available days for the meeting room.
free day (for a person)
Refers to a day when a person has no plans and is available.
Please tell me your free days next week.
Compound of 空き (あき, 'emptiness, vacancy') and 日 (ひ, 'day'), with rendaku voicing ひ → び.