MOSFET to dim LED backlight
Summary: 
It looks as though you can just drive BL directly with the PWM-output.
What happens when you do?



Details:

(1)  As shown, even  if the circuit did what you wanted, all you would be doing is providing a rail to rail (0V to 3.3V ) signal to the BL pin. You can do this by connecting PWM-out to BL directly.
Does this work?

If the LCD and Arduino used differing power supplies the MOSFET may be useful.

(2) To use circuit almost as-is you need a PChannel FET. 
Swap Drain and Source and it will provide PWM to the BL pin. When BL is high the FET is off. When BL is low the FET is on. The FET used must turn on with about 3V between gate and source.

(3) To use the N Channel BS170 you must drive the gate +ve above the source by an amount that is enough to just turn it one "plus a little more". This is called Vth (Vthreshold). The BS170 has Vth ~= 1.8V at low currents. Vth varies somewhat between samples. When you connect the gate to +3.3V the FET tried to turn on but as the source rises it reduced Vgs until it reaches about Vth. Raising the source above this level reduced Vgs to below Vth so the FET is not turned on.
ie this is what a=accounts for the 1.5V that you see as 3.3V - 1.8V = 1.5V.

To use the BS170,
connect the source to ground,
connect drain via a say 1k resistor to +3.3V,
connect drain to BL,
drive gate with PWM.

BUT why does not just driving BL with PWM do what you want?
A data sheet for the display would be useful.