I play Tide Turner to choose Irelia to swap places. Irelia gets +1 for being chosen and then I pay a power with Irelia legend to ready her. Does she get an additional +1 might from being chosen from the Irelia legend?

No, she does not get an additional +1 Might for being "chosen" by the Irelia Legend (Blade Dancer). She gets a total of +2 Might from the full sequence — +1 from Tideturner choosing her, and +1 from being readied by the Legend ability.

Irelia's own trigger reads "When you choose or ready me, give me +1 Might this turn." The Blade Dancer Legend reads "When you choose a friendly unit, you may exhaust me and pay [A] to ready it." The Legend's ability does not choose or target Irelia — it's a triggered response that activates because Tideturner already chose her, and then it readies "it" (the already-chosen unit). Since no new "choose" occurs from the Legend, Irelia's "choose" trigger doesn't fire a second time. She does, however, get +1 from the "ready" portion of her ability when the Legend readies her. (FAQ #8590, #6528)