Another vote for adding a large capacitor to the output of your regulator to supply the fast transient current demand when the LED comes on. Choose a type that has low ESR / ESL such as a ceramic in the 1-10 uF range. Leave the 0.1 where it is. You can prove if it's current induced noise or a software problem by either removing the LED, or making it's series resistor about 10X larger temporarily. Are you seeing the noise on the Power pin of the PIC too?