1.) Giving your team "fun" names is usually unique to support teams. You'll have to think about how that would make your team feel. Would they feel like part of the org? Would they feel different in a positive way, or would they feel looked down upon?
2.) What would be the purpose of the names? How would this relate to your customer base? e.g. a customer base for a fun product might react positively to "Customer Happiness Hero", but a law firm might not appreciate that as much
3.) What are the general tones of your interactions? If customers are contacting your org about minor issues fun names wouldn't be a big deal, but someone who's entire live server went down and is trying to fix it might appreciate communicating with someone with a more professional title
4.) What effect will your name have on hiring? Studies have shown that certain words like Rockstar and Ninja tend to turn away women from applying to your organization. Do some research on the name(s) you want to pick and see if there's any information on it.
5.) Lastly, does it truly reflect the culture of your organization? You can have a super serious company doing super serious things but if a genuine caring / goofy / silly culture is pervasive throughout, then perhaps a name for your customer support team that reflects that is totally appropriate.