The reason is that disk0 contains your actual drive. If you look there, you see that only 125.4 GB of space is allocated to the "Core Storage". Disk 1 is the logical volume (Macintosh HD) which resides in disk0s2 (also named Macintosh HD) so deleting disk1 won't be what you want to accomplish and the tool is preventing you from deleting that "logical device".
You can fix this with the cs subcommand of diskutil in terminal to remove the core storage and put things back on disk0s2 in a none-CoreStorage format.
For safety's sake, make a good backup before continuing and then revert the core storage volume:
diskutil cs revert 8F5A679C-06AB-44CC-BD46-33E327446D80
Now let the machine revert to none-CoreStorage layout. Once that's done, you can use the Disk Utility and or diskutil to resize disk0s2 to take up all the space. Then you would re-enable file vault if you want core storage going forward. This may take more time than wiping the drive and reinstalling the OS and restoring your backup - but you will learn more by using diskutil and not need to test your backup now if you've never restored it.
I personally don't see any benefit to using Core Storage unless it's for encryption, but you're free to do so. Also, there is an undocumented diskutil cs resizeStack operation that I'd be hesitant to rely upon, but if you don't care to lose the data, might be a one trick pony to get your resize done.
I'd just revert the core storage to normal storage and then grow your normal partition using Disk Utility from reading the various comments and your initial post closely.
I decided to streamline the partitioning scheme as much as possible. I deleted my leading, unused partition, but this still didn't make it go.
Somehow the partition type of the EFI partition had gotten changed, so I decided to fix that. There doesn't seem to be any obvious way to do this from within OSX, so I rebooted off of a live Ubuntu boot image on a USB drive. I used gdisk to set the type of the EFI partition to EF, and confirmed that this had also correctly changed its GUID to the correct one for that partition type.
For some reason, under linux the names of the EFI and Recovery partitions had disappeared, so I set them back to the names shown above. I also confirmed that there was a FAT32 filesystem on the EFI partition. Apparently the EFI filesystem is technically a different one from FAT32 but I left it as it was.
And it worked! When I booted back to OSX, I ran Verify on the disk in Disk Utility, which recommended that I run Repair due to some "corruption" in the Recovery Partition, or perhaps just with its header. I ran Repair, which reported success, and was then able to initiate full-disk FileVault encryption.
Best Answer
As per this Apple support question:
So, you have to reformat the entire drive. Remember THIS WILL DELETE ALL OF THE DATA ON THAT DRIVE, so make sure you move all of your files off before you do this. Let me know if this answer wasn't clear or you want help walking through the steps to do this.