I've had the opposite reasoning while working on my project a few months ago.
I'll sum it up here in case its of any help:

I've built beings that are not Gods, but that humans refer to as Gods.
My conclusions are that:
- As long as enough people believe you are a God, you 'could be'
- You should be responsible for the creation of something, or believed to have created something or alternatively, should be the incarnation of something.
Basically, whether you did or not do something is the difference between a God and a False God.

Being the incarnation of something means that you are the paragon/best example of that specific thing. The God of War, for example, should be a master of war, which comes with distinct traits:
- Superhuman strength
- Cunning Strategy
- A will to fight (break peace in favor of war)
All of this can quickly relate to personality traits. You'd imagine he has a short temper when it comes to trivial matters, but that he is more thought when it comes to fighting. Though once he has seen his opponent's flaw, he exploits it without mercy.

I think working with personified Gods would be more fun, because, since they are not the incarnation of something, there is much more design space for their personality and gameplay.
One could argue the Christian God could be something like Zeus, yet, another could perceive it altogether different both in strengths and physical means. The themes surrounding the Christian Gods are vast and vague. Being heavenly could relate to controlling the winds and thunder, or being thought, yet, he is said to command rains of flames and armies of believers on the earth.

I understand my thought process is disorganised, but make whatever you can of it

Immortal?

Gods are thought up by men, and look alike them. They are on a much larger epic scale, but they do the same thing.
Imagine a strong wizard or warrior, and make everything more 'epic' if that's what you're after.
That's pretty much what they made in God of War ;)

A feeling of otherness, to me. In a strategy game, you're probably focusing on destroying or conquering, so those are obvious choices. Most gods in fiction (and many in history I'm sure) had the ability to cause massive death and destruction through one means or another.

There is the opposite route too though: healing, resurrection, curing of disease, blessing crops to sustain populations, and so on.

Generally, gods can do things that regular people can't. Their knowledge and wisdom are usually beyond mortal reasoning. They are often eternal or without need for sustenance, and their care for mortal worship is rarely explained.

So, what makes me feel like a god? It depends on the game and world a lot, but in general it's the ability to do powerful things when the time is right. If I'm playing a personified god of war, then I want to be able to crush my foes personally in massive numbers. I want to inspire allies around me and invoke fear in my foes. If I'm playing an impersonal god of war, things are different. Perhaps he focuses on bolstering the strength and resolve of his allies while doing the opposite in his foes? Maybe he disrupts the tactics of his foes to give his allies an advantage.

That's all the action bits. What a god does outside of battle varies so tremendously that it's hard to pin down what else a god would be expected to do. Some might expect worship or sacrifices, others might not care at all. Some may dispense punishments for going against their rules, some may reward for following them, and some may have no rules at all.

I'd say the ability to lay down tenets of worship, dole out reward and punishment, and gaining strength and power through worship are a good start for a traditional feeling god. It might go the opposite direction though, maybe the god is initially created by the desires of their subjects and are shaped by them. Maybe the god is slave to these desires, or perhaps is then able to direct the growth of their nation.

You could basically make it a Civilization game where the god even directs the entire future of their empire. How they factor into it personally though is another story.

Nietzsche said that the only power humans had over Gods (which could kill them) would be to stop believing in them. To him, God died (read, Christianity) because people stop believing that their God was the reason for all moral things. Obviously, he wasn't being literal, but it can be applied.
Thus, a God is only as powerful as the amount of followers he has.
There would be designspace for an RTS there, obviously, where followers and faith are resources employed by the Gods to maintain themselves and be powerful.
I can see fun in trying to break my opponent's faith or followers as a suitable alternative to sheer power. Opens up 'intelligent' gameplay as an option on the margin of raw power.
That said, I'd pick a theme that is as far as possible from current religions to avoid bad opinions (they don't make good publicity generally).