The way you were testing the current (amps) is not exactly a good indicator of your battery's actual power output, but more for testing the internal resistance of your battery. Instead you should have the meter in series with your load/light to get an accurate current/amp reading. The meter is also showing what appears to be 1mA not 1A. When you tested the voltage you are seeing 5v on an open circuit (no load) and that voltage will drop significantly once you add one (light/phone/etc). The best way to know your power output is to test voltage (at the load/light/etc) and current (meter in series with load) simultaneously, then just multiply the two to get watts. So 5v x 1A would be 5 watts, 5v x 1ma (or .001A) would be .005 watts. Once you add a load that exceeds the output of your power supply (earth battery) the voltage will drop so that little current will flow. You will probably need a much bigger setup and/or many more earth batteries wired in combinations of parallel and series to get maximum current + volts in order to get up to 1+A @ 5v for charging devices. Look up how wiring multiple batteries in different configurations can help you get the voltage/current you are looking for. Also, if you want more lights illuminated at the same time remember (when wired in series) that the voltage divides between each load depending on each load's resistance so if you have 2 - 3 volt lights wired in series you will need 6v+ to give each light 3v. If you wire them in parallel you will only need 3v but your current requirement will double since the resistance will be cut in half.
V=I•R
Voltage = I (current in amps) x Resistance (ohms)