I think the best fit for a near-future postapocolyptic setting would be cryptocurrencies.  

So your funds would be determined by unutilized computing power.  Essentially it'd mean an oppositional relationship between research (using your computing cycles to solve real problems) and money (using your computing cycles to solve meaningless problems that make you money).

How would cryptocurrencies translate to ability to construct buildings and machines of war? It seems more like something you use to fool someone for real currency and then use real currency to buy resources/items in a working economy.

It's all equally imaginary; cryptocurrencies can be the underpinning of a working economy just as well as pieces of paper or shiny metals.  Why will Boeing make me an airplane if I give it lots of pieces of paper?  Because their supplies accept it, their employees accept it, and on down the line, and the employees accept it because grocery stores and banks accept it...  What keeps it going?  A shared understanding that the pieces of paper are "worth something", combined with a government decree that businesses have to accept them.

The benefits of deciding that crypto underpins the economy in your game world is that it's immediately empire-wide by nature (like you don't have to transport it), and it's produced by buildings you already have (power plants and/or research centers).  It's basically a way of theming "energy" as "funds", without having to worry about how you actually transport "energy" across the map.

On the note of Crypto Currency. It doesn't really add anything then?

Back to the original question:

You can have:

• Mining
• Agriculture
• Forestry
• Services

The idea of services is that they can be spent to maintain equipment(ie heal), new research or make your people happy(more produce). It is an abstraction of the service industry today.

Not a bad idea (service).

How about forestry (or maybe "logging", forestry seems to be more focus on caring for the forest, which doesnt suit a militaristic dictatorship maybe. But the world "forestry" fits the modern theme better i think). Does it seem it's ok for a modern "income source"? I know it's a big deal in todays economy but i think it's more related to consumer products, not so much military. But it's still used (wood products) in construction of course...

Logging/forestry would be agood to squeesz in as in "funds income source" becouse i will most likely skip "power resource" and power plants (too many different gameplay elements).

I would go for:

Money (funds) - this represents labour, it's used for most things. Generated by cities.

Resources (ore/minerals) - this represents rare substances required for some advanced units (infantry and simplier units require money only).

Fuel - fuel for mechanized units, limits the number of heavy units

As for science and the like I would go for a secondary resource. Like, you can set how much money goes into science.