Using a compartator to make an LED light up. How do I make the LED light up?
From the datasheet: "The outputs are n-channel open-drain configurations".  This means that the when the comparator is "on" (In+ > In-) then the comparator output pin is shorted to ground, but when it is "off" (In+ &lt; In-) the output pin is high impedance (effectively not connected to the rest of the output circuit).

So, you'll want something like your option (2), but without the GND connection to the LED.  When the comparator is on, there will be a path from 5V to ground through the resistor, LED, and output pin.  When the comparator is off, there will be no current path through the LED.