Killzone SF
As Shifty's pointed out, it's really hard to guess. The X1X is capable of delivering native 4K games with higher resolution textures using 12GB of GDDR5. So that, coupled with marketing requirements, indicate that slightly more than 12GB is the bare minimum.
Should the next-gen consoles contain an NVME SSD, it could fill up 16GB of memory in 1 second. As Shifty said, the amount of memory required for buffering could reduce substantially.
Should they contain a secondary pool of memory for UI and apps, that will increase the amount of main memory accessible to games. For example, let's say they cut OS requirements (and I mean OS, not UI) down to 1GB, whilst increasing main memory to 16GB. Although only double the size, we'd see nearly triple the capacity.
Bandwidth seems to be the greatest concern, and 24GB of GDDR6 on a 384bit bus is the best, cheapest way of meeting that requisite bandwidth. Let's be kind of conservative, and assume 1.5GB needs to be reserved for the OS - apps and UI dwelling in secondary memory. That's four times the PS4's available capacity, and approximately four times the bandwidth, depending on the speed of GDDR6.
What I'd love to see though - and I know this is highly unlikely - is two stacks of HBM3, totalling 24GB and 1TB/s. Plenty of bandwidth and enough capacity to last a good 8-10 years. Also, it'd use less power than the same capacity GDDR6, leaving power budget to be spent on higher clocks for the GPU/CPU.
There's a veeeeeeery slender chance we'll get that if there's any truth to the claim that HBM is the future because GDDR is beginning to hit its limits. I know nothing about the veracity of that claim.