**EDIT:** *Ok, the OP is assuring us that this issue started a couple of weeks ago and that that this isn't an April fool's joke.* 

In which case, I would still do this:

Stay your ground. Do not change your name. Send out an email requesting that they implement the same solution that they used for your coworkers with apostrophes in their name. Be a broken record about this. 

But at the same time, be open to the possibility that your situation is different than theirs. Perhaps, your name is significantly longer than theirs. Or perhaps, there is a some hidden field that you're not aware of. Or perhaps, the others were able to change their active directory name without messing up their "display name" in other parts of the system. So be sure to phrase your email in such a way that allows them to come back to you and explain what's going on. 

> Part of me wants to tell them that they must find a work-around,
> because I'm not changing my name. However, another part of me really
> wants to avoid causing drama, when I'm only new in the company.

If you don't like drama, don't change your name. 

If you agree to change your name to suit their system, without knowing why the others didn't need to change theirs, you will regret it. You will resent them for forcing you into that situation. 

And once you do give them your permission, you won't be able to change it back because it will no longer be that IT person's problem, it will be yours now. 

And yes, having the correct name can be important. Not having matching names can screw up employee directory autocomplete searches when people try to get ahold of you. It can screw up reference checks. It could delay bank deposits. It could screw up with some of your benefits/your taxes. It can delay visa applications. It can get you a secondary screening anytime you need to take a plane. And if nothing else, this issue will mean that you'll need to re-introduce yourself every time someone misuses the incorrect name.

And again, I'm not saying you should be categorical about this. I'm only saying that the burden is on them to explain to you why your situation is so different from the other employees (that have apostrophes in their name). The time to ask questions is now, not later. Don't agree to something you do not fully understand. 

  [1]: https://en.wikipedia.org/wiki/April_Fools%27_Day