Does an Irelia Defiant get +1 Might for being chosen from the Reavers Row trigger if she doesn't move back to base?

Yes, Irelia, Fervent will gain +1 Might for being chosen by the Reaver's Row trigger, even if you ultimately decide not to move her when the ability resolves.

According to RiftJudge FAQ #8628:

• Choosing is distinct from resolving: You choose the target (Irelia, Fervent) when you place the Reaver's Row trigger on the chain. Because she has been "chosen," her ability triggers immediately, granting her +1 Might.
• "May" decisions happen at resolution: The decision of whether or not to move the unit to base is an optional ("may") effect that occurs only when the Reaver's Row ability resolves.

Sequence of events:

1. You defend at Reaver's Row, triggering its ability.
2. You place the trigger on the chain and choose Irelia, Fervent as the target.
3. Irelia, Fervent's ability triggers because she was chosen, and she gains +1 Might.
4. When the Reaver's Row ability resolves, you make the "may" decision: you can choose to move her to base, or you can choose not to move her. Regardless of this decision, the +1 Might she gained from being chosen remains for the turn.