The Targaryen girl who foresaw the Doom was Daenys the Dreamer.
However the video, as you described (I haven't watched it), appears to be wrong when they say Targaryens were the last (As in only) Valyrians who escaped the doom.
Targaryens were one of the forty Freeholder families of Valyria all of whom had dragons. When Daenys had a dream about destruction of Valyria, Targaryens did not flee alone. There already were Valyrian Houses Celtigar and Velaryon in the Narrow Sea Isles. Targaryens made Dragonstone their seat while Velaryons settled in Driftmark and Celtigars settled in Claw Isle.
You might find it interesting, While Targaryens lost their possessions after Robert's Rebellion, Velaryons and Celtigars still held theirs. Lord Monford Velaryon was a key vassal of Stannis Baratheon and died fighting for him during Battle of Blackwater bay. His half Brother Aurane Waters fought for Stannis too but changed sides after he was taken PoW and rose to the position of Tommen's Master of Ships. Lord Ardrian Celtigar also fought for Stannis in the battle but changed sides after he was taken PoW.
Furthermore, there was a sizable population of Valyrians in the colonies of Lys, Volantis, Myr, Norvos, Qohor, Lorath, Pentos and Tyrosh. Valyrian blood still runs strong in the Old-Blood of Volantis and Lys.
We can't even say that Targaryens were the last Dragonriders who escaped the Doom. Emperor Aurion, self proclaimed Emperor of Valyria, was a Dragonrider and was in Qohor when the Doom fell. He proclaimed himself the Emperor, raised an army of 30,000 Qohoriks and went to reclaim Valyria with his dragon but he and his army disappeared during their quest. That event made Targaryens the last Dragonlords in the world.
So in conclusion:
- Daenys the Dreamer is the woman in question.
- Targaryens were not the last Valyrians.
- Targaryens were not the last Dragonlords.