What is the branch name variable for Jenkins multibranch pipelines?

The env.BRANCH_NAME variable contains the branch name.

As of Pipeline Groovy Plugin 2.18, you can also just use BRANCH_NAME (env isn't required but still accepted.)

There is not a dedicated variable for this purpose yet (JENKINS-30252). In the meantime you can take advantage of the fact that the subproject name is taken from the branch name, and use

env.JOB_NAME.replaceFirst('.+/', '')

This has now been resolved, see Krzysztof Krasoń's answer.