1989 Ford F-250 Lariat won't start. Please help!

1989 Ford F250 Lariat, either a 7.3 or 7.5 L engine. Has always started reliably but was dead when I went to start it. Put charger on battery overnight and it started fine. Started two more times and then had to be jumped to start. Battery gauge needle wavering back and forth in a weird way. Friend borrowed it to get to work, but when he got in to head home it wouldn't start, so we figured it was the battery. Purchased a new battery and installed, but still won't start. No noise at all when turn the key. Rapped on starter a couple of times and still nothing. Read up on it and now figure it may be either the ignition, or more likely the solenoid. Need help to diagnose. Truck is inaccessible to me as it is parked in a Boeing lot that requires security access and only my friend can get access. Can't afford to tow it, so need detailed instructions as to what he can try. Thanks.