Whenever I want to do any type of hardware things on any Mac I have owned, I go to
www.ifixit.com. Looking at that site for the 13" Macbook A1278 (that A1278 seems to be the designation for your machine), it looks like you can replace the hard drive:
https://www.ifixit.com/Device/MacBook_Unibody_Model_A1278
https://www.ifixit.com/Guide/MacBook Unibody Model A1278 Hard Drive Replacement/816
(The second link is actually from the first link, which shows all the various types of repairs you can do.)
Now, I see where the original hard drive inside your machine was 256 gig in size. When you replace the drive, what do you plan on doing with the "old" 256 gig drive? One reason I ask is that maybe you can do what I did with the original (slow) 1 TB drive I had inside my Mac Mini:
1. Purchase a 256 gig SSD (I bought a Samsung 840 Pro; excellent drive, and fast).
2. Purchase a small, slim 2.5" external case.
3. Install the SSD inside that external case.
4. Install Mavericks on it (your machine can run Mavericks; by the way, the last update to Mavericks, 10.9.5, came out yesterday).
5. Remove the original 256 gig drive from your Macbook, and remove the 256 gig SSD drive from the external case.
6. Install the 256 gig SSD in your MacBook, using the instructions from ifixit.
7. Boot the Macbook, and hopefully it will work. You'll also appreciate how much faster it boots up.
8. Install the original 256 gig drive inside the external case. You can then connect it to your Macbook.
I did exactly the same thing for my Mac Mini, except the original 1 TB (slow) drive went into a slim, external case.
I guess the other thing you need to decide is to whether a total of 512 gig is enough for your needs. Myself, beyond the OS, I do not have too many other things on my SSD (also have a 250 gig SSD inside my 13" Mac Book Air). I just try and manage it the best way I can. By doing that, I am not even using half the storage space on either SSD (nice to have the extra space, though, just in case I download some large files). If necessary, you could purchase a larger, 2.5 " drive and put it into the external case. Then, you can try and sell the original 256 gig drive (after completely wiping it clean). Or, you can use that original 2.5" drive in another external case for backing up the information from the SSD inside the Macbook machine.