Extracting water from the martian landscape would literally be seen as a source of wealth, and it gives people a strong incentive to extract as much water from the landscape as is reasonably doable, thus giving society a steady source of water, which drives society forward.
Maybe, 1 martian dollar ($1) would represent 1 ton of water, and 1 cent (or “bit”) would represent 1 kg of water, which means 1000 cents/bits = 1 dollar.
So you might have paper money representing 5c (almost a gallon) or 50c (let’s call it a barrel) for everyday use, so when you go to a market to buy some food, you might pay for it either by paper money or by literally handing over water bottles.
The paper money would have to be handed out by banks that store the water or at least are able to hand it out if somebody wants to redeem their paper money.
And to follow up on my own comment, imagine this:
For the love of god, don’t call something a cent (1/100) if it is worth 1/1000
xD it’s to confuse the romans, in case they invade again