Power Supply Problem with Arduino Yún: Using a log USB cable
The USB specification limits the length of a cable between two devices to 5 meters. When you make a longer cable, you're going out of spec, which means it is not guaranteed to work anymore.

Since you're only supplying power, the issue is (as you have guessed) the voltage drop in the cable. However, the voltage drop will depend on how much current Arduino is consuming at any given moment. Supplying it with a higher voltage without a converter is dangerous, because it will lead to overvoltage you won't be able to control.

I'd suggest trying better cables first: thicker and stiffer cables have thicker wires, and may have less voltage drop. You can also make your own cable (provided you can solder) using only two thick wires for power.

The ultimate solution, like you said yourself, is to deliver higher voltage over the wire (12V would do) and install a power converted (12V to 5V) near your Arduino. That way, the voltage drop in 5V line will be small, and the voltage drop in 10m 12V line will be compensated by the converter. Actually, the intermediate voltage may be higher: you can simply drop a mains line (220V or whatever you have) to your weather station and place the same USB charger near the Arduino.