An application runs on multiple EC2 instances behind an ELB.

The solution that will meet the requirements is to write data to Amazon ElastiCache. This way, the application can write session data to a fast, scalable, and reliable in-memory data store that can be served reliably across multiple requests. The other options either involve writing data to persistent storage, which is slower and more expensive than in-memory storage, or writing data to the root filesystem, which is not shared among multiple EC2 instances.