This process has been such a pain in the ass, so hopefully this will save you some stress.
The HP Split has two hard drives. A small SSD (64gb) with the OS in the tablet portion and a 500gb hard drive in the dock. I wanted to upgrade the SSD in the tablet with a larger one. I also didn't want to reinstall the OS or the many programs I had on this computer. I bought a Samsung 840Evo 250gb mSATA SSD for a very reasonable price off Amazon. I also bought a USB 3.0 mSATA bay so I could clone the drive (even though this computer doesn't support USB 3.0, I'll use it later). I backed up all of the files first, then started the process.
First problem: The OS is Windows 8. I never got around to upgrading it before. I tried 4 or 5 times to get Windows 8.1 to complete the upgrade, but it never would. I tried just about everything I could find online: did all updates (OS and drivers), uninstalled antivirus software, etc. I eventually gave up on this. I'll try again one more time after I get the new SSD installed.
Second problem: The new drive (in the USB bay) wasn't recognized in windows explorer. You have to go into disk management and format the drive, then it will show up.
Third problem: Samsung has free migration software available on their website. Turns out it is total shit. It failed 1/4 the way through cloning. I found the EaseUS Todo Backup Free software online that seems to be fairly adware free. I installed it and cloned the C: partition. I then switched the SSD's.
Fourth problem: It wouldn't boot after the switch. Shit. I switched them again, windows ran a system recovery, and I got back to where I started. I reformatted the new SSD. I then read a bit more. Turns out you might need to clone the whole drive including the tiny partitions, so I did that.
Fifth problem: That didn't work and now the original SSD won't boot at all. It says I need recovery media, which of course I don't have. So I've actually gone backwards. Damn it.
I did a factory reset of the original SSD. This required hours of installing updates. I was then able to update to Windows 8.1, which means some piece of software was preventing me from updating the drive to Windows 8.1 earlier. I then installed missing drivers from HP, set a system restore point, installed EaseUS Todo Backup Free again, and attempted another clone to the new SSD. I first tried booting off of the new drive while it was still in the USB bay, but that didn't work. The computer restarted and booted off the original drive. At least the original drive didn't fail this time. Next, I just replaced the original drive with the new drive and unplugged the USB bay to see if the computer was getting confused by having two identical copies of Windows. It booted up! YAY! I had to resize the volume of the partition under disk management to make full use of the volume. Now I have a hybrid laptop with ~230gb SSD and a 500gb HD!
So, steps you should follow if you want to do something similar:
1. Backup your files and programs.
2. Set a system restore point.
3. Reinstall the OS (I did a factory reset). I couldn't update to Windows 8.1 nor make a successful clone due to some (unknown) software preventing it from working.
4. Perform all software and driver updates
5. Set a system restore point
6. Clone the whole drive with windows on it. I suggest staying away from manufacturer provided migration software. There are a lot of good, free cloning software out there.
7. Swap the cloned drive with the old drive.
9. Install programs and put your backed up files back on it.
Total time spent upgrading to a new SSD: 12 hours