Its pretty normal for people to act differently around different people. But that doesn't seem like what is happening here. This is a delicate situation indeed. Your friend must get the point across, "I have to walk on egg shells around you", without offending or hurting the feelings of the person and making them feel like it is their fault (even though it kinda is).
The hardest part about having a conversation like this is making sure that the other person isn't put on the defensive, because then the exchange of meaning is completely shut down. Once a person feels backed into a position where they must defend themselves the chances of coming to a healthy agreement drop essentially to zero. Much of my advice from here on out comes from the book Crucial Conversations, which I would highly recommend for you friend to check out in-depth.
- Start with heart
Focus on the goal of the conversation. If, but probably when, things become heated stop and reassess your goals. When we get emotional we start to lose track of what we were in the conversation to accomplish in the first place.
Refuse the "fool's choice". You don't have to choose between peace and honesty, these can come hand-in-hand as long as the conversation stays rational. When they don't, focus on what you don't want -- You don't want an argument. You don't want to break up. You don't want to hurt them.
- Learn to Look
Keep monitoring the content and conditions of the conversation. Are we focusing on what is crucial, or are we arguing about irrelevant minutiae, keeping us off-track? Be mindful of signs that your partner is moving towards "silence or violence" -- are they shutting down or getting defensive? Are you moving towards an outburst? Its good to try to assess for symptoms of your own irrationality so that you can identify and realign when they occur in the conversation.
- Make it safe
When your partner is becoming emotional you start to lose the mutual ground that allows the conversation to happen. Is the tide of the conversation risking your mutual respect or mutual purpose? Contrast to fix misunderstandings -- start with what you didn't intend then explain what you did intend. Brainstorm together for solutions.
- Master your stories
Just as important as trying to keep your partner from moving away from rational discourse is making sure you stay on track as well. When angered or hurt, retrace your path. Notice your behavior, what emotions are at play, what "story" is creating those emotions in you? Is that story logically sound and derived from factual evidence? If not, you may be misinterpreting. Always analyze the evidence. Ask yourself questions, "am I pretending not to notice my role in this?", "why would a reasonable person do what I am doing?", "What would I do right now if I really wanted these results?
Share your facts. Tell your story (your point of view). Ask for other paths. Talk tentatively, be flexible. Encourage opposing views, let your partner air out their concerns.
- Explore your partners "path"
Restore conversational "safety" with curiosity. Ask for their views. Acknowledge their emotions. Paraphrase to show you understand and to demonstrate that they are safe to open up to you. When in doubt, take a best guess at their thoughts, but dont accuse. Make a point to express and recognize when you agree. Build from that agreement. Compare differing views.
- Move to action
Finish clearly. Determine who does what, and when. Make each others' expectations crystal clear. Hold yourself accountable to promises you make.
Your friend might also want to be open to the possibility that this just might not work out. This was a big enough deal for her to feel she had to change her personality around him to accommodate his sensitivities. What happens if the person isn't willing to deal with their "true", more playful personality? Going back to hiding that part of herself isn't a healthy and sustainable option. Just something to consider.