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See instructions here for creating a bootable El Capitan USB drive. When you boot from that, erase the drive with Disk Utility in order to do a clean install.

The best way to backup would probably be Time Machine, but simply copying the Home folder and other files works as well if she doesn't need to keep any apps.
 
On the assumption you're yet to remove the old drive:
  • Backup Downloads, Documents, Music, Photos and Desktop
  • Replace drive
  • Boot to Internet Recovery
  • Install the available version of OS X
  • Upgrade to latest version of OS X if necessary
In the worst case scenario it all goes pear shaped, you can replace the old drive to still have a known working installation.
 
When I did this, I used one of the disk cloning utilities (super duper) to clone my drive to the SSD by attaching it to the USB port first (will require something like an external enclosure). Then I swapped the drives. Gave me a 'disaster recovery' backup drive with the old drive, which I now keep updated regularly, but that is assuming you don't want to use it for something else.
 
My suggestions:

1. Buy a 2.5 external USB3 enclosure (as well as a new SSD). One like this will do fine:
https://www.amazon.com/Support-ORIC...8-2-fkmr2&keywords=inateck+sata+2.5+usb3+uasp

2. Put the SSD into the external enclosure for now.

3. Initialize the SSD with Disk Utility.

4. Install a clean copy of El Capitan to the SSD.

5. Connect the SSD to the sister's MBP (if it's not already connected). Now boot the new copy of El Capitan for the first time.

6. Go through the initial setup procedure. The setup assistant should offer you the option to "bring over" from an older drive your accounts, apps, settings, data.

7. Do so. When all is done, REBOOT from the external SSD. Take A GOOD LOOK AROUND to see that things are as you wish them to be.

8. If everything looks good, NOW is the time to open up the MBP and "do the drive swap". BE SURE TO USE THE RIGHT TOOLS. You can find the "how to" guide at ifixit.com.

9. Take the old HDD from the MPB, and put it into the external enclosure. It can now become a backup drive, supplemental storage, etc.
 
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Also, if her machine has Mavericks installed, and I boot into internet recovery, will it install the latest El Capitan or will it download Mavericks?
It will download whatever her computer shipped with. For example, the original owner of my MacBook Pro bought it in early 2013 so it downloads Mountain Lion when I do Internet Recovery.
 
Do you have another mac around?
Download El Cap on that one and then use a 8+GB USB to make a bootable USB installer using diskmaker x http://diskmakerx.com
then you simply insert the new SSD in you sisters mac, and boot holding option down, select the Installer on the usb -> disk utility to formate the SSD into the right system and then you go back to the menu and select install os x
 
Do you have a 8 GB USB thumb drive or larger that you can erase and format if so:

1.) Download El Capitan app from the Apple App Store for free.

2.) Format aforementioned USB drive as a journaled partition with the GUID option using Mac OS X's built in disk utility. Make sure you name the volume MyVolume (no space capital M and capital V).

3.) Open native Mac OS X terminal.app and copy and paste the command below verbatim. You will need to have administrator rights on the computer and enter your password. 20 short minutes later you'll have a universal El Capitan USB installer you can use an any modern Mac (capable of running the new OS).

sudo /Applications/Install\ OS\ X\ El\ Capitan.app/Contents/Resources/createinstallmedia --volume /Volumes/MyVolume --applicationpath /Applications/Install\ OS\ X\ El\ Capitan.app

To boot from USB device press and hold the "option" key right after you hear the Mac "chime" as it's booting up.

https://support.apple.com/en-us/HT201372
 
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