If I Flurry of Blades on a battlefield during a showdown, and a 2 might unit lives with 1 damage taken, can I assign only one damage from my unit to kill it or do I need to assign the 2?

Yes, you can assign only 1 damage to kill that unit.

According to Rule 460.2.c.2 and the clarification provided by RiftJudge FAQ #871, damage that is already marked on a unit persists and counts towards determining if a unit has received lethal damage.

Because your unit with 1 damage marked on it has 2 Might, it requires a total of 2 damage to be killed. Since it already has 1 damage marked on it from your Flurry of Blades, you only need to assign 1 additional point of combat damage to reach that total of 2. Once the total damage assigned and marked equals or exceeds the unit's Might, it is considered lethal, and you are not required to assign any further damage to it.