Are you sure the voltage output varies with the solar panel? It could be set up so the voltage is constant and amperage varies.
Do you have a volt meter to test what kind of voltage you are actually getting?
If you have tested and know the output voltage varies and your voltage is in the 3 volt range then -
Are you sure you are attaching positive and negative connectors properly to the LED? An LED only works one direction. What is the voltage requirement on your LED? A 12v LED won't light with only 3 volts. Too much voltage on a 3v LED will burn it out.
What kind of amperage does your solar panel put out? If it doesn't supply enough amps it probably won't operate toys and appliances that require more.