Is there a simple hash function that one can compute without a computer?
Well, depending on what's reasonable and what you can compute reliably by hand, you MIGHT be able to compute modular exponents in a moderate-size group (like $Z_n$ for moderate $n$).  Exponentiation via repeated squaring is doable I think.

Probably more practical would be a MD5-like ad hoc confusion-diffusion primitive.  You'd need to make the input large enough to make exhaustive search impractical by hand (32 bits should do it I would think?) and then make a round-function that was simple enough to compute by hand but had empirically-good collision-resistance.  Finally apply Merkle-Damgaard for a "few" rounds.

I would expect this to work reasonable well with a random crypto classmate.  If you're playing against Adi Shamir, you're probably screwed.

Edit: For in-person commitment schemes, you can get rid of hash functions altogether.  Just write a string of random bits on a folded paper and hand it to your partner to commit those bits.  But the more interesting scenario is doing this through a prison wall, over a phone, or over the internet... then you really need something mathematical. 

Let's take the prison-wall example: you want to commit to another inmate some integer $x$ and you have no computational resources but can communicate with tapping (no paper).  Here's what I would do: pick a 64-bit number $x$ that you want to commit.  Make sure you can't factor it (it might be prime, but verifying primality is going to be tedious by hand).  Then choose another, larger, 64-bit number $y$ that is also hard to factor by hand.

Now compute $p=xy$ (easy) and send $p$ to your buddy.  If $x$ and $y$ were both hard to factor, $p$ is as well, so he can't extract $x$ or $y$ from $p$.  Now for the reveal, you send $x$ and $y$; he verifies that $p = xy$ and that $x$ is the smaller integer.

It's important that both are 64-bits (or both the same size) to prevent you from changing to $x&#39;$ after the commit where $p = x&#39; (k y)$.  This doesn't happen if $x$ and $y$ are prime, of course, but large primes aren't easy to generate by hand.