The starting material for convenient preparation of deuterated hydrogen peroxide (D$$_2$$O$$_2$$) in laboratory is:

We need to identify the starting material for the convenient laboratory preparation of deuterated hydrogen peroxide (D$$_2$$O$$_2$$).

Analysis of each option:

Option A: K$$_2$$S$$_2$$O$$_8$$ (Potassium persulfate)

The hydrolysis of potassium persulfate with heavy water (D$$_2$$O) gives D$$_2$$O$$_2$$:

In this method, deuterium comes from D$$_2$$O (heavy water), which is readily available. This is the most convenient laboratory method for preparing D$$_2$$O$$_2$$.

Option B: 2-ethylanthraquinol

This is used in the industrial anthraquinone auto-oxidation process for large-scale H$$_2$$O$$_2$$ production. It requires catalytic hydrogenation using D$$_2$$ gas, which is not practical for routine laboratory preparation of D$$_2$$O$$_2$$.

Option C: BaO$$_2$$ (Barium peroxide)

The Merck process uses BaO$$_2$$ with dilute acid: BaO$$_2$$ + H$$_2$$SO$$_4$$ → BaSO$$_4$$ + H$$_2$$O$$_2$$. For D$$_2$$O$$_2$$, one would need D$$_2$$SO$$_4$$ (deuterated sulfuric acid), which is expensive and not conveniently available. This makes it impractical for D$$_2$$O$$_2$$ preparation.

Option D: BaO (Barium oxide)

Barium oxide is a basic oxide, not a peroxide. It cannot directly produce H$$_2$$O$$_2$$ or D$$_2$$O$$_2$$. The reaction BaO + H$$_2$$O → Ba(OH)$$_2$$ only gives a hydroxide.

The correct answer is Option A: K$$_2$$S$$_2$$O$$_8$$.