But Verin very clearly wanted to pick Egwene. She could have gone to Siuan, to the head of the Brown Ajah, the head of the Green Ajah... any number of people, including Cadsuane, who is definitely the most competent Aes Sedai around.

She picked Egwene, and her reasoning is clear to see. Egwene has been working against the Black for a time now, and by all means, was the single person who was most successful in frustrating their plans. As Verin said, Egwene's actions within the Tower had really frustrated the plans of the Blacks. She also seemed well on her way to being undisputed leader of the AS, which made her perfect for giving this data to. No one else could have managed to kill one third of the Blacks in one stroke and drive all the others away. Elayne and Nynaeve would have done the same in her position, but they were busy elsewhere.

The reason Egwene was the only choice was because she had made the most consistent effort to reunite the Tower and foil the Black's plans. Which meant Verin could happily give the names to her and trust completely that she'd do exactly the right thing, which she did.