Current booster for relay
You can't in the way you try it. Your source is 250mW, your relay requires 500mW. You have not got enough energy.

A transistor can amplify current but not out of thin air. It needs to come from somewhere. 

What you can try is to see what the relay hold current is. It is often much lower then the 'attack'* or "pick" current. In that cause you might store energy in a capacitor and use that to get the relay to 'start' en then your 50mA may be enough to 'hold' it. 

*I can't remember if that is the right term...