The developers of Multibit have explicitly disabled such a feature to change the transaction fee, citing concerns that having little or very low transaction fees would result in the transaction taking too long to process (which is going to be unbearable for the average user only interested in sending and receiving money quickly).

From the GitHub issue originally submitted, the developer explains:

The reason for this is that whilst in theory it is possible to send BTC with zero fee in practice it is not. For the "regular user" (i.e. someone who just wants to use bitcoin and does not really know the details) their user experience is quite poor with zero fee tx. We have probably both experienced this with Instawallet (zero fees added) where you sometimes have to wait until the next morning before the miners decide to add your tx to a block.

It would create too much frustration to users (and too many support calls for me) to have zero fees.

Note: tx means a transaction.

As of Multibit Classic 0.5.19 (last in the series) you can now do this. However, the max transaction fee ratio is too low for today's market and miners! If you want to spend out of this client, even to upgrade to another client, you'll need to do something like this:

Compile a modified client

1. Clone the repo: git clone git@github.com:keepkey/multibit.git multibit-classic; cd multibit-classic
2. Check out the most recently released version: git checkout v0.5.19
3. Edit the max transaction fee in src/main/java/org/multibit/viewsystem/swing/view/components/FeeSlider.java from 50000 satoshis to something larger, e.g. 100000 (i.e. 1 mBTC).
4. Compile with Maven: mvn -DskipTests=true package
5. The executable is target/multibit-exe.jar

Use the modified client

What I did was install v0.5.19 the usual way, then swap out the multibit-exe.jar in the install directory for the modified one I made. You can probably just run the jar file directly, but I can't test that out at the moment.