It’s not unusual for a series to have characters that are mysterious, but many times, at least the characters themselves know their own past lives. That said, in sci-fi and fantasy, there’s also the added potential mystery where a character doesn’t know anything about his or her own past, and there’s a sense of discovery involved when done right.
Odo doesn’t really know anything about where he came from.
What little both the show and Odo seem to know is about him at this point is the Cardassians found him on an asteroid and more or less made him what he is: the single-minded head of security for Deep Space Nine. He’s a dogged investigator, honest and incorruptible to a fault, and largely uninterested in socializing. He needs to be in charge of security at all times, and he does tend to have a superior attitude towards, if not everybody, then at least to those who break the law. Arguably, the only person he shows any respect for is Kira though he has some kind of relationship with Quark, one where he looks down his nose at the Ferengi but likewise he seems to spend a lot of time in Quark’s bar talking to Quark.
I think we could make a case where Quark and Odo are more or less friends. At the least, they have a friendly adversarial relationship that I think both characters secretly enjoy.
As it is, Odo knows nothing about his species or where he came from. And that could change when he arrests a fellow for murder. Said fellow comes from the Gamma Quadrant and calls Odo a “changeling”. He also has a crystal of some kind which, when tested by Bashir, shows a genetic structure similar to Odo’s own.
That said, how Odo arrested the guy was clever. Quark was having a backroom deal with some raiders, twin brothers of Miradorn race both of whom were played by the same actor. Rom brought in some drinks on a tray, and Odo was hiding there as a drinking glass. Quark isn’t dumb either and threw out enough lines to suggest ignorance to keep himself out of trouble. Granted, one of the MIradorns were killed, and the surviving twin said in his race that was like losing a piece of himself and he was going to need to get the Gamma Quadrant Guy.
That said, the man knew next to nothing, only that changelings are said to exist by his own people. He does get Odo to detour into a vortex that allegedly has other changelings in it, but it was really a stasis pod for the man’s young daughter. And with the Miradorn firing on them, Odo and the guy need to work together first to get the girl out of the pod (the crystal shapeshifted into a key), and then get both back to the runabout.
Interesting facts: Odo can be knocked out by a rock in his humanoid form. You’d think he’d just splat or something. Likewise, he’s a lot heavier than he looks.
Now, after rescuing Odo, Odo does the same by setting up the Miradron to blow himself up, then asking a passing Vulcan science vessel to take the guy and his daughter away, promising to uncharacteristically tell a bunch of lies to cover for the man who, even though he’s a murderer, did save Odo’s life, and the murder was something of an accident.
So, Odo is still mostly a mystery, but now we know he’ll bend the rules a bit if someone can give him some answers.