The flight departs at 7 am for international connection in London, and I would like to check in my bag the night before because I'm elderly and there will be no one to help me get my bag down three flights of stairs during my early morning trip to the airport.
While your airline will definitely let you check in online for your flight up to 24 hours before the time of departure, it's relatively unlikely that they will let you drop your bags with them until the flight checkin desks at the airport opens, 3 to 4 hours before your flight. The only exceptions appear to be at Gatwick and Edinburgh, which do allow evening check ins for early flights. You could try contacting them with a plea for special assistance, but I'm not going to guarantee you'll get anywhere.
On the other hand, Geneva airport does offer some paid for luggage services, with self service lockers available from 03.30 to 0.30 each day on the railway station level. There is also a staffed counter, but that doesn't seem to open early enough for your purposes.