How would I replicate this speckled/pixelated background effect?
You might need to zoom in to see the effect fully.
This is pretty simple.
Create your background layer and fill it with the color that you want. Then choose Filters>Noise>Add Noise... from the menus. Make sure Monochromatic is checked, and adjust the Amount until it looks how you want.
Hope this helps!
This is the way which i know....
From very starting first of all
In this you can reduce
This is my output...
I have used this size to make you understand you can apply this on very small size also....
you can follow screenshot as they are to get the same result as you want..
You can go with Eric Thayne answer also, its pretty easy and clean but i doubt it matches or not with your requirements...
Hope this will help!!