What is the most secure hand cipher?
Solitaire by Bruce schneier is probably your best bet.  It has a few issues but it will work well for most things.  It ends up having a small bias, but it takes about 15 seconds per character after the initial key stream has been generated.

It is not nearly as widely studied a field as most people are assumed to be able to get a computer or other computation device.  The only other one I can think of is VIC.  It is based on a lagged Fibonacci generator which is a bit of a different place to start from. See this other answer for more details on that cipher.

I'd strongly suggest looking long and hard at Solitaire. It has really a cool design.