If I have Nami and she holds. When I play Irelia, it readies her and buffs her. Does she get a +1 for being chosen?

To answer your question, we must look at whether Nami's "Hold" ability "chooses" or "targets" Irelia, Fervent.

Short Answer: No, Irelia, Fervent does not gain an additional +1 Might for being "chosen" when Nami's ability readies and buffs her. She only gains the +1 Might from the "ready" trigger.

Reasoning

According to Rule 355.10, a game object is only a "target" (and thus "chosen") if it is explicitly targeted by a spell, activated ability, or triggered ability.

1. Nami's Ability is Non-Targeting: Nami's ability reads: "When I hold, the next time you play a unit this turn, ready it and [Buff] it." This effect is applied programmatically to the unit you play; it does not require you to "choose" or "target" that unit.
2. Irelia's Trigger Requirements: Irelia, Fervent’s ability states: "When you choose or ready me, give me +1 [S] this turn."
   • Since Nami's effect does not "choose" Irelia, the "choose" portion of Irelia's ability is not triggered.
   • Because Nami's effect does "ready" Irelia, the "ready" portion of Irelia's ability is triggered, granting her +1 Might.

Sequence of Events

1. Nami Holds: Nami's ability triggers, creating a lasting effect for the next unit you play.
2. Play Irelia: You play Irelia, Fervent.
3. Replacement/Modification: Nami's effect applies, readying Irelia and buffing her.
4. Irelia Triggers: Because Irelia was "readied," her triggered ability is added to the chain.
5. Resolution: Irelia's ability resolves, giving her +1 Might.

Because Nami's ability does not target or choose the unit, the "choose" trigger condition for Irelia is never met. (This is consistent with RiftJudge FAQ #6528 and #8590, which clarify that effects which apply to a unit without targeting or choosing it do not trigger Irelia's "choose" bonus.)