I used this weed barrier under my Mexican River Rocks, and now weeds are growing through the barrier.
Any recommendations for what kind of material I can use to prevent this from happening again?
It might have "failed." It probably didn't.
In most cases, this will be new weeds having started in the new dirt and organic debris that accumulated around your rocks on top of the (unfailed) weed-block fabric.
As such, the only way this sort of thing "works" long term (even if you pour concrete under it) is if you periodically remove and wash the rock and remove all accumulated dirt and organic debris.
Sounds like (and is) a huge hassle. But if it's exposed to the world, it will inevitably collect dirt and organic debris, and seeds will sprout there.
If weeds are in fact growing through the barrier, you need to buy a better quality barrier. Product recommendations are specifically off-topic, so you'll have to figure that out on your own.
I would take a chemical approach. You really don't have any other choices at this point. You could maintain a routine to spray out individual weeds, but that's a lot of work. If it were me, I'd apply a long term vegetation killer to the entire area. There are many available, some are per-emergents like Preene. Others like Casaron, Ortho Ground Clear Freehand (if available where you live), Crossbow, Roundup. etc . Product recommendations are OT here so I'm not making a recommendation, just listing a few. You'll need to research "long lasting vegetation killer" . You may be opposed to a chemical approach, or it may not be allowed in your area. But I don't see many alternatives other than spot treatment and a lot of pulling weeds.