Some thoughts:
The Crucial SSD you described should work fine.
IMPORTANT IMPORTANT IMPORTANT
BE SURE you have THE RIGHT TOOLS for the replacement job.
I believe you'll need a Phillips #00 screwdriver and a TORX T-6 driver. (these worked on my 2010-vintage MBPro, but check ifixit to be sure).
These can be found at hardware stores, Home Depot, Lowe's, etc.
Review the ifixit.com drive replacement guide for instructions, if you've never opened up the MacBook before.
If you have the replacement drive and enclosure "in hand" before you visit her, I would advise you to do a "prep and test" on the drive beforehand. Put it into the enclosure you have and install the OS of your choice onto it.
Then, boot to the new install by invoking the startup manager at reboot (hold down option key until SM appears, then select the external drive).
At this point things can get tricky, depending on what you want to do.
If the new drive boots to the point where it asks you to begin the initial setup, you could just cancel out of it (even power down if necessary), and "leave it be" until you get the new drive installed into the MacBook.
The idea here is that you know the drive will boot, but you want your sister (and the old drive) to be present for the setup, with the possibility that the old drive MIGHT connect and mount, and offer you the opportunity to migrate her old data to the new drive.
That's one way to do it.
Another way would be to create a "temp account" on the new drive for now. You might just use "administrator" for the username and a simple password.
The idea here is that when you arrive at your sister's, you can install the drive and boot it right up -- and then try to connect the old drive to see if it's possible to migrate old data over to the new one.
If you go this route, once you get the new drive installed, you might try migration assistant first. It might be able to bring over everything -- her old account, apps, data, and settings. If this works, you can set things up so that it boots right to her account (instead of "administrator").
Be aware that if this doesn't work (or if the old drive still gives you mounting problems), it -IS- possible to do a "manual migration", including items that are inside the "old home folder".
BUT -- very important:
You CANNOT simply copy the main sub-folders from one home folder to another. By "main sub-folders", I mean the folders named "Documents", "Downloads", "Mail", "Movies", "Music", "Pictures", etc.
HOWEVER -- you CAN copy all the items INSIDE those folders, from the old folder to the new folder.
By doing this, it's possible to "fairly re-create" the contents that were in the old home folder, in the new home folder.
If the old drive still gives you problems, there are other ways to "get at" the data.
But try to get the above done first.
The key here is, get the new drive installed, and get it bootable.
Then, worry about what's next....