It's a Realtek RTL8188E. I found a bug-report on it, basically saying it isn't supported by Ubuntu. There is a fix-- Linux kernel 3.10 and up supports it, so I will have to manually upgrade my kernel to the stable 3.10.
If I disable secure boot then Mint boots up. If it is enabled, then Windows 8 boots.
I installed rEFUnd, but the laptop UEFI ignores it. If I replace the boot_x86.efi file under /efi/EFI/Boot directory, then secure boot has a conniption (if enabled).
I am thinking that there is a battle of bootloaders going on here and that Toshiba is rigging the fight. There is a /toshiba folder in there, so I may try putting the rEFInd bootloader in that directory, replacing one or both of the ones in there.