Since he mentions sponge, I don't think he's talking about a mute switch, but rather the foam mutes people put under the bridge cover to damp sustain. Not what we normally think of as an "effect", but what else is it really.
To the OP, the budget-minded are usually satisfied with a bit of sponge or some rolled-up old sock; while the champagne-minded use a bassmute.com mute. I had one of those bassmute ones, and it's really well-made.
No need for a special material. Sponge will be annoying to use as soon as it dries. Any soft cloth will do the trick, just stick it under the strings at the bridge. A sock, a hanky, whatever.
I use the same cloth I wipe my strings with.
