Here's what I whipped up today:
Despite the poor quality of these photos, primarily I'm a photographer and I have lots of "7.4V" Li-Ion batteries for my cameras. I went to the local retail camera store and spent AU$10 on a charging plate (for their generic Li-Ion charger) to match the Canon BP-511/A batteries I have. I then opened it up (takes a screwdriver), replaced the internal cables (takes a soldering iron, but it's just wires: no electronics), and reshaped the back of the plate with a knife.
The plug was cut from the end of the AC adaptor that came with the H4 (no point in keeping it as it only handles 120V AC which we don't use around here).
The battery clips into the plate, and the whole thing is held onto the back of the H4's tripod mount by the mount's velcro strap. The strap fits snugly along the back of the battery (you might notice in the first photo that there's a despression along there between two bulges).
I'll probably replace the DC connector with a lower-profile right-angle connector later, and am looking at putting a low-voltage warning LED into the battery mount. For now I'll just cycle fresh batteries into it without letting them run too low.
As for power, the batteries are typically rated at 1100-1600 mAh, with a nominal voltage of 7.4V (although I measure over 8V).
A 7.4V 1400 mAh battery theoretically has twice the capacity of a pair of 2300 mAh AA NiMH batteries but only about 2/3rds the capacity of the 9.6V 1600mAh battery shown at the start of this thread (although the voltage/current/time behaviour of Li-Ion vs NiMH means that the comparison isn't that simple).
So the run-time will be longer than with the internal batteries, although obviously there's another benefit: the hiss and beeping produced by internal batteries with an un-modded H4 are completely gone.