This is actually something that a turing machine is suitable for.

Get youself something to remember what state you're in, write the rules on a sheet of paper and use a stack of paper as the tape, with each piece of paper being one cell.

Turing's idea was to build a mathematical model that mirrored how humans go about large, complex tasks.

Honestly, all of these super basic mathematical models for calculation are fairly well suited to being "run" by hand.

Random-access-machines can be done similarly to turing machines. Write the program down, keep the program counter on one piece of paper, and have a (numbered) stack of paper as memory.

And lambda-calculus is probably the easiest. Just write out the expression, and then write out each reduction step. This is what functional programming languages build on, and with the additional effort of checking types, you could probabl evaluate something like Haskell by hand pretty well.

All of those are turing complete (if you ignore infinite memory).

EDIT: spelling