Best is subjective. Here's my subjective approach, so caveat emptor, take what I say as personal belief and not a professional benchmark testing...
For AOSP-based roms, here's what my experience is:
Lean: Easiest and simplest of the kernels. Has limited governors, but the tweaked InteractiveX is pretty effective at doing what you need. It also has a crapton of S3 specific tweaks to improve battery life. You can overclock to 1.72 (I believe...) Downside? The min frequencies are 384k, which means other kernels can go into a deeper sleep. The TCP algorythms are limited too, so choose 'reno'.
DKP: I love this kernel, but several people report it doesn't work for them. This kernel has a ton of governors, with 'freelunch' and 'asswax' being the best. There are also a lot of TCP algorythms (choose 'yeah' for best performance). You can overclock to something ridiculous like 2.1... but I keep it at 1.62; there's also deep sleep frequencies down to 54k (yes... 54000), but I tend to stay at108k. The downside? Some people can't get this kernel to work on their device and it doesn't have some of the battery saving tweaks Lean does.
KT: This is kind of the 'go to' for many people, because it has a standalone app that flashes in the process and the dev is pretty active with the S3 version here. Many of the same features of the previous two, but different governors and nothing that's a device-specific tweak; the kernel is developed for multiple devices.
BMS: This, to me, is a similar kernel to KT and was my go-to for 4.3 roms. I haven't used it in a while, so I can't speak to if it offers more governors and customization.
My take and choice?
Lean and DKP offer the best battery and performance than the rest; they seem to give the same performance, so really you can use either and be happy. I find KT and BMS to be sluggish (even with tweaking) on 4.4 roms AND they don't seem to put the device into the sleep I want it to for battery savings.
BUT... If you're flashing Gummy, give the stock kernel a try! It's the only AOSP-based rom that doesn't use the Cyanogenmod kernel and you can actually get some good performance out of it. I flash DKP over Gummy, but I also flash Lean some times.