Oops. yes you're right, the display does turn on and stay on for a minute (which is my timeout). II gave up on solar charging until I got a beefier solar cell.

The problem with the 1W supply is that it can't keep the voltage up at 5V, and can barely keep the voltage above the voltage needed to charge the battery. Ideally you need 2.5W or so to emulate a regular USB port, and if you can do the full 4+W of a wall charger, that would be great. Otherwise it would be best to take the battery out of the phone and charge directly from the solar cell to the battery - that way you don't have to deal with charger inefficiencies and parasitic draw. But yes theoretically at 1W you should be able to charge a BL-5J in just under 1 day if you have no parasitic draw and no conversion inefficiencies.

The DC-DC converter I was using with my solar cell was only around 70% efficient too, I think, so down to 0.7W at the USB port.