We know many of the Time Lords by titles, like The Second, and, of course, The Doctor. There's also Time Lords we know by name, like Lady Romana and Rassilon. We also know of some where we know their names and their titles, like The Master (Koschei) or The Monk (Mortimus).
Why do some Time Lords use names and some use titles? And why, when other Time Lords have freely given their names (even ones that use titles), is the Doctor unwilling to tell his name to anyone other than River Song?
Here's my take after 30 years of watching Doctor Who. This isn't based on any single particular source and based mostly on the TV shows and movie.
First, it's a personal choice on the part of the Time Lord, which can change during their various life spans. Time Lord culture seems to expect this as a normal occurrence. For example, see how Drax reacts when he meets the Doctor in "The Armageddon Factor".
Second, what we hear as humans isn't necessarily what the Time Lords are actually saying. Time Lord communication is at least partially telepathic and possibly machine enhanced. Our semi-evolved simian brains are making the best of the data a Time Lord brain sends to us. For example, see how Sarah Jane Smith questions how she can understand every alien she encounters exactly as though they were speaking English.
Third, using Romana's name as a example, Time Lord personal names seem to be very complicated and long, so they shorten them or replace them with titles for us simians to use.
Fourth, out-of-universe it allows authors to create any kind of Time Lord they need for a story and immediately give them something instantly identifiable and iconographic.
Fifth, (extra point shot attempt) "Doctor Who?" was the punchline of many of the early episodes. When the Doctor left the final scene, one of the characters remaining would say something like: "He was the Doctor", and another would ask "The Doctor? Doctor Who?" Asking if anyone had actually heard his name.
As we've found out, the Doctor's name is the Answer That Must Never Be Spoken, To the Question that's been hidden since the Dawn of Time, Hidden in plain Sight. (which might refer to the Doctor's name written in High Galifreyan on his crib.) Apparantly the Universe itself might just fall apart if his name is ever spoken. The whole reason that the Silence was trying to kill him is apparantly to prevent this Answer being spoken in the one place where all questions must be answered, where Lies and Silence are not permitted. A place linked to the Fall of the Eleventh.
Is that enigmatic enough for you?
Which might indicate that the Doctor himself is something other than just a mad Timelord. He may either be, or connected to a being with far greater scope or antiquity than we know.
Maybe he's just cautious about giving it to humans, who have a short life span. At Trenzalore, the Time Lords asked for his name, because they knew only he would know it. It's an.... inside joke, kind of, except not a joke.
In "Turn Left", Rose said, in effect, that the wrong word in the wrong place, if you've crossed many universes, can tear the universe apart. Maybe it's dangerous to leave too much information about yourself scattered through time.
Another interesting pattern seems to be that runaway or rebellious Time Lords, like the Doctor and the Master, among others, go by titles, while normal Time Lords, like Rassilon, use their real names.
I think it’s to keep them grounded. As we often see when the doctor regenerates they become a completely different person. Their whole personality changes. Imagine if that happened to you, all the music you love, food you like, types of people you get on with... all of a sudden you hate it all. Imagine that. It would tear you apart. 10th said it was akin to dying and you can see why.
But naming yourself after an ideal? That’s powerful. A code you strive for, even when you become a different person. It would help you keep some of yourself, your core personality.
In the doctors travels, he recreated the universe with his mind, what should happen, could happen, and could never be. His name is and should be hidden at the beginning of the universe hidden in the Medusa cascade, so when his name is spoken, the causal nexus and time vortex for the universe will be torn apart and can alter everyone and everything in that universe. This is also how he can go within his own timeline, because the universe can't and won't see him die. Yet everything ends, ....everything.
My personal theory was after the timewar he used his name as a password on the timelock keeping the time war contained. Before that it he just preferred his title for personal and cultural reasons. Another poster does mention the Medusa Cascade. Note the big deal he made about his secret in the Akhaten speech and - to me - implying he rebooted a post timewar universe. By the end of the season though it fizzled out into a secret incarnation (so what?). (Moffat a few times seemed to start a season with some theme and then by the middle of it switches.) Note also the Doctor speaks old Gallifreyan and that is alleged to have some powers. Consider also "Skasis Paradigm" from school reunion that demonstrates that reality can be easily manipulated by someone with the right codes.
The reason he has kept his name secret is because as others have suggested, names have power and they can use his name to get to the other Time Lords or worse, time lock is a definite problem if broken. In my opinion if they learn his name they could prevent him from rebuilding the universe.
