Gaining control of a creature while an Aura targeting it is on the stack
First, if Boonweaver Giant's triggered ability is on the stack, then Boonweaver himself is not on the stack. He has already entered the battlefield. Second, Spectra Ward should never have been placed on the stack. Boonweaver Giant's triggered ability is what actually gets placed on the stack. Your opponent doesn't have to make a decision until it resolves.

With these corrections in mind, here is what your stack actually looks like:


Hypnotic Siren (TOP)
Boonweaver Giant's triggered ability (BOTTOM)


Hypnotic Siren resolves. You gain control of Boonweaver Giant. Next, Boonweaver Giant's ability resolves. Your opponent may search for an aura and attach it to Boonweaver Giant. He may choose not to. If he does, he will control the aura attached to Boonweaver, and you will control Boonweaver.