Well, the exact powers of a child in mythology would vary, but likely it would be centered around Agriculture, quite possibly going into the Harvest and The Seasons.
Demeter is a Goddess, an Olympian, a daughter of Kronos and Rhea. She is by no means a demigod.
There is no way of knowing that.
She had an affair with the demigod Percy
A demigod child of Ares would have skilled control of weapons (without training), superiour horsemanship, and in many cases are master strategists.
When a god marries a demigod the demigod receives the same powers as the god. Which makes the demigod now more powerful the god. So the demigod has their powers and the powers of the god.
The Power Of Zeus Children is the power to throw lighting bolts.
The Greek goddess Artemis was a virgin goddess, so she had no children. If she did have a demigod child, her/his powers would be similar to Artemis'. They would probable be able to shoot an arrow with perfect aim and heal.
Asclepius was really a demigod, but became the god of healing.
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Demeter was born from the Titan gods Rhea and Cronus, she was born a Greek goddess, and so always had her "powers".
She had none, she was a virgin goddess.
She is an Goddess. Of course she has powers.