Why is it that Oberyn went back to Obara's mother and why was he so sure that she was his child?
The details aren’t exactly clear, but it seems likely that Obara was conceived when Oberyn was at the Citadel, around age 13 (271-272 AC). Note Oberyn was also in Oldtown after age 16, during his “exile” for killing Lord Edgar Yronwood, and may have attended the Citadel again (for the first time?) during that period, but either way he acquired six maester’s links at some point. And also either way, Obara was 28 when Oberyn was 41, so, yeah, he was young when she was born – but it’s not uncommon for noble boys of that age in Westeros to have sexual experiences, especially with prostitutes (which Oldtown is famous for). (For example, Tyrion considered hiring one of Chataya’s girls for 13-year-old Joffrey.)
It is probable that either when Obara was born, or when Oberyn returned to Oldtown, the prostitute informed him that she’d had his child, likely for money for support and/or for more of his attention. Whether or not she was absolutely certain he was the father, note that the name “Obara” is certainly meant as a flattery of Oberyn, like Barra the baby born to the Chataya’s girl that Robert Baratheon deflowered.
However, at some point Oberyn got into a responsibility thing, where he wasn’t just going to provide monetary support for his bastard daughters, but raise them himself, including training them in the use of weapons. Obara was about 7-8 when he returned to claim her (based on the fact that she says she was at the Water Gardens 20 years ago and wasn’t there long), which was before Robert’s Rebellion, so it wasn’t an Elia thing, note. Well, not an Elia’s death thing at least, but maybe he was inspired by Rhaenys’s birth, who knows. Or maybe it was Tyene’s birth (around 277 AC) that got Oberyn to start collecting his daughters, as her mother was a septa and therefore couldn’t keep her. And maybe it was Sarella’s birth (around 280 AC) that got Oberyn to return to Oldtown (as Sarella loves Oldtown and Summer Islander ships often come to that port, it’s possible she was born there).
Whichever. Some time around 280 AC, Oberyn returned to Oldtown, to the prostitute who had borne him a daughter, and demanded that Obara leave with him. Which the woman did not take well:
“The day my father came to claim me, my mother did not wish for me to
go. ‘She is a girl,’ she said, ‘and I do not think that she is yours. I
had a thousand other men.’ He tossed his spear at my feet and
gave my mother the back of his hand across the face, so she began to
weep. ‘Girl or boy, we fight our battles,’ he said, ‘but the gods let us
choose our weapons.’ He pointed to the spear, then to my mother’s tears, and I picked up the spear. ‘I told you she was mine,’ my father said, and took me. My mother drank herself to death within the year. They say that she was weeping as she died.”
–Obara Sand, AFFC, The Captain of Guards
So, despite her mother’s desperate and hollow excuses, the reasons Oberyn was sure Obara was his child:
Her name was Obara, named for him
It is probable that her mother said she was his when she was born
She chose the spear as her weapon instead of a woman’s tears
And when Oberyn claimed her, Obara eagerly left with him.
(For the record, Obara hates Oldtown and wants it destroyed. It is extremely probable that as a prostitute’s daughter, she had a terrible childhood. Especially if it was known her father was from Dorne, since Reachmen are horribly racist to the Dornish. Also, at her age, it is extremely probable she was beginning to be trained in the profession, if not worse already. That does not excuse Oberyn’s abuse of her mother, but it could explain why Obara seems to have felt no concern for her or her death.)
Obara, daughter of Prince Oberyn and a Oldtown whore. Brought to Dorne when she was six, Obara is big boned and fierce. She carries a spear, whip, and shield with her. When Elia died, Obara was twelve and had known her aunt for several years. Her father mourned so deeply that it left a lasting impression on her and kindled a hatred of Lannisters.
Nymeria, daughter of Prince Oberyn and a noble of Volantis. She was only two when her mother gave her to her father to care for. Called Lady Nym, she has all of her mother’s beauty and political sense. Her preferred weapons are daggers, and she has been known to hide up to a dozen of them on her person. When Elia died Nymeria was nine, and although she had only met her once she saw the grief it caused her father.
Tyene, daughter of Prince Oberyn and a Lannisport septa. Her mother fled to Dorne when she realized she was pregnant, and although she has since returned to the sept, Tyene has never known anything but Dorne. In her youth she tried to be as devout as her mother, encouraged by her love of embroidery and music, hoping to join her in the sept. When Elia died, Tyene was seven and had never met her aunt. The more she grew the more she heard, and when she set aside her dream of the sept it was because she never wanted to be as helpless as Elia had been. Tyene can brew and use poisons as well as her father, and no matter what blade she chooses, it is poison that is her weapon.
Sarella, daughter of Prince Oberyn and a captain from the Summer Isles. Even as a child Sarella wanted to know everything. She read her uncle’s history books and begged to visit the places she heard about in stories. From the moment she learnt of the Citadel, she wanted to go. Oberyn had never denied his daughters anything.
(Based off their appearance from the books: Obara, Nymeria, Tyene, and Sarella Sand. Each is incredibly distinct, with wildly different skills, appearances, and personalities. They are actual strong female characters, rather than being a squad of stupid, murderous teenagers like in the show.)