A friends Honda Shadow has a weird starting problem

A friend of mine has an early 2000, maybe late 90's Honda Shadow 1100 that's having a weird problem.
It only wants to start when the battery has a full 12.8 volts. He can start it and run it for a while, let it sit over night, and the battery will drop to 12.5 or 12.6 volts, it'll turn and then noth'n.
He said replaced the solenoid and the plugs, and the battery is new. I didn't know what to tell him.

Any thoughts?

Honda eletricals

Some suggestions:
-Start with all the typical possibilities, loose connection at the battery or at the other end of the positive and negative cables. The description almost sounds like that's the problem. One bump and the loose connection arcs a little. That causes a small burn spot which interrupts the connectivity. Also check the battery post adapters if present. Verify the adapter (usually brass) is fully contacting the lead post coming out of the battery. It is possible to use too long a bolt in the top of the post adapter and that can result in less than optimal contact. But be careful not to overtighten.

-The starter button is known to give problems. The contactor builds up dirt and carbon. There's some small parts in it but it can be disassembled and cleaned with some patience.
Here's a write up from the Shadow ACE Tourer forum.
http://hondashadowacetourer.yuku.com/topic/7430/t/Dirty-starter-button-contacts.html

-Shadow charging systems are also problematic. If your friend has added electrical accessories (lightbar, etc) the connector between the alternator and rectifier has been known to overheat. It can become a charred mess of plastic and wire insulation. The connector is under the saddle, typically bundled in the mess or wires just behind the tank in a vinyl "cup" full of wires and connectors. Heck, even without add-ons, this connector has been known to bbq itself.

Check all ground points too. A high resistance path back to the negative post can wreak as much havoc as a bad device and mimic starter, regulator, or solenoid issues.

Just because the battery is "new" doesn't mean it is GOOD! Try subbing a "known good" battery to verify the new one isn't the culprit.