I just run most of the games from a USB stick. I would go with an HDD, I have dozens of ~250 GB laying around, but I don’t have the network/HDD expansion, YET
I have a PS2 slim without the HDD bay and I’ve read that the USB 1.1 is slower than the 10 Mbit/s network connection. A PS2 fat with HDD bay would really be the best option.
I also heard that USB 1.1 is slow, but it seems faster than the DVD drive, and that’s my only comparison point. I was going to get the memory card to SD card adapter since I’ve read is faster, but I don’t care honestly.