If Zhonyas triggers, can I use the Blade Dancer ability?

Whether you can use the Blade Dancer legend ability when Zhonya's Hourglass triggers depends on whether the Zhonya's Hourglass trigger involves "choosing" a friendly unit.

According to Rule 355.10.c, a game object mentioned in the text of an ability is not a target (and therefore is not "chosen") if it is included only as part of a replacement effect.

The Breakdown

1. Zhonya's Hourglass Trigger: The text of Zhonya's is: "If a friendly unit would die, kill this instead. Heal that unit, exhaust it, and recall it." This is a replacement effect that automatically applies to the next qualifying event (the unit dying). Because it is a replacement effect, it does not "choose" or "target" the unit (see Rule 355.10.c).
2. Blade Dancer Trigger: The Blade Dancer ability triggers: "When you choose a friendly unit..."
3. The Conclusion: Because the Zhonya's Hourglass trigger does not "choose" the unit it saves, the Blade Dancer ability will not trigger in response to the Zhonya's effect.

Summary: You cannot use the Blade Dancer ability to ready a unit simply because it was saved by Zhonya's Hourglass, because the Zhonya's effect does not count as "choosing" the unit.

This is my interpretation based on Rule 355.10.c and the mechanics of replacement effects. No official FAQ exists for this specific interaction.