Looking for name recommendations for a project about having great conversations.

I love this question! Gets my creative mind flowing early morning.
Here a few from top of my head:
-Heart2Heart
-Tete (tete a tete reflects a deep personal conversation between 2 people)
-InterCo (for interaction & communcation)
-InAccord (self explanatory)
-tootalk
-chatspot
-lucidity (play on clarifty)

Even if you don't like them, I kind of like some of these. Please feel free to google me, if you feel i can be of any help let me know. I'm one of Arizona's best commercial strategists.

Umberto has some good ones. Here are some more, also off the top of my head. Hard to nail down without knowing the application (naming strategy can be different depending on whether it is a product or service, etc.). Also, I obviously haven't investigated, but depending upon how you plan to use your name(s) and how publicly, you might want to do some basic checks on possible TM conflicts.

2Connect
2Engage
Connexion/Connexions
People First

I would also focus on names around Empathy/Empathic communication, which is really what it is all about.

Happy to help more formally if/when you'd like.

That depends on what is unique and "evergreen" about your approach. A lot of companies exist to encourage conversation in the workplace. I'd focus not on the product's features but on the end result - the benefits, and in particular, the emotional benefits of using the product.

For example, Clarity.fm supports conversations but it's not called "Expert chat" - what a boring brand that would be! Instead they have chosen the territory of "clarity", which is the feeling you WANT to have when you have finished using the product. It's the brand's promise. I am willing to speak with an expert, and to pay them, when what I want above all else is Clarity. I could hunt for answers on the internet, but it's often overwhelming with a lot of conflicting advice. When you speak with an expert however you feel a sense of clarity.

So for you, what is the unique value prop? What is the feeling you are promising to create? Happy to discuss. Calls with me are free right now. Give me a ring and we can brainstorm.

- Caneel Joyce, http://kickassenterprises.com

Just a quick few:
- Emgage (employee engagement)
- Emex (employee exchange)
- OpenEx (Open exchange)

Personally, I own Dialoguing.com, which sounds like a natural fit. But I'd be happy to work with you on name creation and domain procurement, both of which are areas I work in professionally.