What version of InDesign are you using? What is the source application for your data (Excel, a database)? Are you sure that those are just single apostrophes?
I just did a test in InDesign CC 9.2 and exported a .xls file as a .csv file. I changed a name to "O'Connor". I had no problem previewing or creating a merge file with that name.
Are you maybe working on a Mac with a CSV generated by an Excel user on Windows?
When hooking up your CSV, look for the "Show Import Options" button. There you can select your Delimiter (usually a comma, right?) as well as your Platform and Encoding. If my guess above is correct, then you want to set it to ASCII and PC but if it's some other arrangement, you have to either figure out the correct platform and encoding by trial and error, or figure out the encoding and platform source of the CSV by other means.
By the way, David Blatner has just released a great new title at Lynda.com about Data Merge.
Here's a link to the announcement:
Here's a link to a weekly trial of Lynda.com for free:
Thank you for the suggestions! Turns out (I'm using InDesign CC 9.2 and was creating the CSV from Numbers) that I needed to use Unicode (UTF-16) instead of UTF-8 to get it to show up properly.
In fact, David mentions using Unicode (UTF-16) in his videos.