As it stands, all demi-gods are all more or less beings with extraordinary stat-caps....and pretty much just that.
But when I think of the tales of old, these demi-gods, like the deity that sired them, all had a particular theme. Hercules was strong enough to hold up the sky. Achilles and indeed many other "demi-gods" from various cultures, such as Sigurd or Esfandyar were invulnerable in all but a single point. An achilles heel if you will.
SO WHAT EXACTLY DO I PROPOSE?
When generating a demi-god, the most important factor is where the divine parantage resides. So, let's just say the game pics a god at random, and comes up with one of your deities that is associated with the sky and fire. It would then randomly choose sky or fire. And alter your demi-god accordingly.
Let's say it chose fire, and then decided to slap the Lava-Immune tag on your character. He could then swim as he likes ((provided he knows how to swim)) within lava. His only concerns that his attire would burn away to nothing and leave him naked. Or maybe drowning.
SO IN A NUTSHELL. Demi-god is gifted with a random power, randomly chosen from what is associated with his randomly picked Divine parent. ((and since gods are randomly generated as well...that's just even better.))
Powers could range from the incredibly useful : Can fly, Can breath water, Giant Strength ((ie. treated as a giant in mass for wrestling/striking/related things)), doubling your speed from what it already is, having unbreakable bones. Never sleeps, Eternally young, Etc.
To the less so. : Generates Miasma, Always Drunk, Blood/Vomit burns or freezes things, Immune to all forms of syndrome, inspires tantrums. Can mine without a pick. Etc.
And maybe, every so often generate a weakness to go with a demigod. Like weakness to a certain material ((anything, from silver to larch, would be insanely DF to have a legendary demi-god killed with a larch figurine of himself)), abilities that only manifest during night or day, or maybe even only when holding a certain object, like a flute or short spear. Sky is the limit really.
AND LET'S NOT FORGET
You really...-really- should set it up to generate demi-gods in world gen, and then again throughout history. And don't just limit them to humans/dwarves either. I can already imagine a demi-god bronze collossus that breathes fire, whorshipped by a cult of goblins.