It’s an interesting question, famously posed by the great Philip K Dick. But despite a novel and a movie it is not easily answered. And certainly not with a circle of leather and a web of woven string.
Capturing modern dreams calls for modern thinking and modern designs. To this end I have created an Android’s Dream Catcher shown below.
Its wireless interface maintains a constant connection between their central processing unit and the SkynetCloud™ during their REM (Resting Electronic Mode) periods.
It is much more advanced than the analogue model upon which it is based:
Stannous Flouride's Book of Truthiness 