Why would you say "je vais me faire l'avocat du diable" instead of "je vais faire l'avocat du diable"?
"Se faire" can have two meanings.
The first one is used in a sentence where someone does something to the subject.
Paul se fait battre par John.
Paul is being beaten by John.
In this case, Paul is the subject of the sentence, but he is the one who is receiving the beating. In can be compared to an English Present Continuous passive sentence (how twisted!), the passive of "John is beating Paul".
But there is a particular meaning when the subject of the sentence is the one who does the action.
Je me fais petit pour éviter d'être vu.
I make myself small to avoid being seen.
In this case, "se faire" means "to make oneself" or "to turn oneself into". Literally or figuratively. You can say "La chenille se fait papillon" ("the caterpillar turns itself into a butterfly").
Back to your example:
Je fais l'avocat du diable.
I do/play the devil's advocate.
Je me fais l'avocat du diable.
I make myself the devil's advocate.
Both are correct and used.
As the writing is similar, we usually don't see the difference in French, especially when speaking. But "I make myself the devil's advocate" implies that you are trying to look like one, even if you are not, pretty much like "I make myself big" of "I make myself strong".
Don't confuse that with "Je vais me faire un café" ("I will make myself a coffee"), which is a third meaning that I shall not discuss here, for it has a completely different context.
Also, don't confuse that with "Je vais me faire cet abruti", which is slang for "I'm going to beat up that moron". The meaning of "me faire" in that example is similar to "make myself a coffee", except it would be the moron's face you "prepare to your own taste".
Les deux peuvent se dire mais le rôle du locuteur est différent à chaque fois.
Je me fais l'avocat du diable (emploi de la forme pronominale (se + V) du verbe faire) : j'ai décidé que je serais l'avocat du diable, c'est ma décision.
Je fais l'avocat du diable : ça peut-être ma décision, ou alors on me l'a imposé (jeu de rôle, débat fictif, etc....)
We can say both, but they each have a different meaning.
Je me fais l'avocat du diable (reflexive verb) : it's my decision, I have decided I'd be the devil's advocate.
Je fais l'avocat du diable : it can me my own decision or it has been set upon me (for instance in a roleplay)