Why do lights get brighter when I turn on the hair dryer?
I didn't think of this but adding to Spehro's addition to KingDuken's comment this might help. The socket might be on L1 and the LED light on L2.







simulate this circuit &ndash; Schematic created using CircuitLab

Figure 1. A North American split-phase domestic supply.


Let's say L1 is 120 V and your hairdryer is about 10 Ω so 10 A will flow. 
Let's say there is a (very high) neutral resistance of 1 Ω then there will be a 10 V drop across R2. 
Now L1-N = 110 V and L2-N = 130 V so the lights will get brighter.


You should be able to prove this to yourself by monitoring the lighting circuit voltage with a multimeter while switching the hairdryer on and off.