How do you get the internal drive to show up when you're reinstalling?
The comments about encryption pointed me in the right direction.

Can you reinstall OS X on a FileVault 2-encrypted drive? has the answer; you need to erase the drive first, and you need to do it from a terminal.

Use diskutil cs list to get the UUID of the Logical Volume Group. It's the first listed:

    CoreStorage logical volume groups (1 found)
    |
    +-- Logical Volume Group BE3F9C36-A105-4854-9A5B-79511B0303B5
        =========================================================
        Name:         System
        Status:       Online
        Size:         67859718144 B (67.9 GB)
        Free Space:   12656640 B (12.7 MB)
        |
        +-&lt; Physical Volume 9BB1A946-B429-47D0-970B-99CD78D8E502
        |   ----------------------------------------------------
        |   Index:    0
        |   Disk:     disk1s2
        |   Status:   Online
        |   Size:     67859718144 B (67.9 GB)
        |
        +-&gt; Logical Volume Family F3AB68B3-6831-4BEA-B509-B1BF97DFC65E
            ----------------------------------------------------------
            Encryption Status:       Unlocked
            Encryption Type:         AES-XTS
            Conversion Status:       Complete
            Conversion Direction:    -none-
            Has Encrypted Extents:   Yes
            Fully Secure:            Yes
            Passphrase Required:     Yes
            |
            +-&gt; Logical Volume 4B5B7F22-9113-4813-B5B6-FF1C1117F0B9
                ---------------------------------------------------
                Disk:                  disk2
                Status:                Online
                Size (Total):          67494739968 B (67.5 GB)
                Conversion Progress:   -none-
                Revertible:            Yes (unlock and decryption required)
                LV Name:               System
                Volume Name:           System
                Content Hint:          Apple_HFS


then diskutil cs delete UUID

With the above example: diskutil cs delete BE3F9C36-A105-4854-9A5B-79511B0303B5

(If you're looking at the startup menu that has four options - Restore from Time Machine backup, Reinstall OS X, Get Help Online, Disk Utility - you're in the right place.  Terminal isn't one of the obvious options, it's on the Utilities menu at the top of the screen.)