The Xbox 360 cannot format USB flash drives. You must format the drive on a computer first—specifically to FAT32, the only filesystem the 360 recognizes. Here's the foolproof process.
Note on Drives >32GB: Windows' built-in formatter refuses FAT32 on drives larger than 32GB. Use a third-party tool like GUIFormat (fat32format), Rufus, or EaseUS Partition Master Free to format 64GB, 128GB, or 256GB drives to FAT32. The Xbox 360 supports up to 2TB externally (with latest dashboard), but 16GB-64GB is the sweet spot for reliability.
Format on PC to FAT32. Plug into 360. Done.