He Calls Himself Pride
As Solas is not Dalish or city elf, he has no clan name to speak of—so Solas is the only name we know him by. In elven, "Solas" means pride, and through companion dialogues and other events in the game, we can see his name is fitting. Unlike previous pages in this section, I can't avoid talking spoilers without leaving the essential discussion of his name out, so take that as your last warning.
When we meet him, his own introduction comprised almost entirely of him saying his name (he spends the rest of his dialogue emphasizing the destructive aftermath caused by the Breach). After your meeting with him, the next time you open up your game journal and view your Codex entries, you'll find his entry is no longer unknown. His Codex entry is the note Leliana (one of the Inquisition's advisers) wrote to your other companion Cassandra Pentaghast. Leliana describes how Solas joined the Inquisition, and gives some of Solas's back story. Her words reflect a natural amount of apprehension at an unknown apostate mage offering such openhanded help.
This apprehension dissipates once Solas has proved himself to be extremely useful to the Inquisition again and again (with his knowledge of the Fade and finding Skyhold as a new base). As we get to know him, he reveals through companion dialogues that in his youth he was once "hot-blooded" and "cocky," fitting in to his prideful name. In more cryptic dialogues with Cole, your spirit companion, Solas reveals he has much regret from mistakes made in his youth.
From these we can see, while Solas may not be full of pride now, he was once such a character. Post reveal it's not entirely certain if Solas was ever his real name. In the post-epilogue version of his Codex entry, Leliana laments her trust in his story and not investigating it further while the Inquisition was so busy. As names go, I think it's safe to say "Solas" was most likely a real name, or at least one of his real names, since despite his subterfuge he has been pretty consistent with his story.
"Voice ringing with fullness from both worlds, guiding me to the shining places. He calls himself Pride."
When you confront him in Trespasser, he states "The Dread Wolf" is what his friends and enemies called him, as his power and reputation grew, but that "Solas" came first. Knowing his name means "pride" will make some secondary dialogues more meaningful, such as when Cole mysteriously talks about someone calling himself pride (his interactions with Cole are always the most illuminating).