Mystique/Foxx text says:

Pay 2 endurance, reveal a card from your hand: Choose a name or affiliation in that card´s name, affiliation, or game text. Mystique gains that name or affiliation during the combat phase this turn.

Suppose I give her the name Mr. Siniestro and play on her Malleable DNA, that says:

Target Mr. Sinister you control has invulnerability and can´t be the target of opposing effects this turn.

When the recovery phase starts, is Mystique still invulnerable? Or as she lost the name she additionally lost the power?

Thanks.

Although Mystique's ability only lasts for the combat phase, Malleable DNA only checks that it is a Mr. Sinister when it tries to resolve. Once it resolves, it is for the rest of the turn, even the recovery phase.

That's my opinion. I'm not 100% sure.

I agree. That said, there's a chance that it does a 'check again' thing like Carrying the Torch does.

What is the interaction you're trying to make? Are you trying to do something to stun her again or something?

I don't know the interaction, it's a question posted in the spanish forums.
But I suppose he will try to stun it for some reason during the recovery.

When an effect uses the word "target," you check the legality of that target exactly twice. Once when the effect is being played and the target is first chosen, and a second time right before resolving it. After the effect has resolved, the targeting requirements no longer matter.

Case in point, look at cards like Malice 6-drop---"Target stunned character can't ready this turn." By definition, stunned characters are never ready; this card targets a stunned character, but it clearly continues to apply after they've recovered. Otherwise it would do nothing!