I've actually given this a considerable amount of thought myself, although I would steer away from the simple house. Where's the fun in that?
The factors of how much a house would cost in game are much like the way you would do it in real life. We must first identify;
- Area. Location, location, location!
- Readily available resources, will likely be dictated by (1). If you build a house in the desert, you wont likely be building a house of wood, eh?
- Economy; are we building in a fairly rich area or is the local economy peasant bartering? Assuming we are in a fairly stable market using gold coins, what's a valuable item as compared to a invaluable one?
- Housing market; a subset of economy. Are there existing houses in the area or not? Is the land your building on rich in resources or is it barren? Who owns the land? Who rules the area? Is there a high demand for housing and low supply, or vice versa?
- Lastly we have to consider living standards. Does your player's character just want a clean little log cabin and won't mind a leaky roof? Does she want to furnish the building with only the finest furniture? How much does a kitchen matter to her? Indoor plumbing? Is indoor plumbing even a thing in your world?
Once we work out the general ideas (at least) for these questions we can begin looking at the ultimate equation:
We know that the fighter in question has an elevated standard of living, therefore she will more than likely want to live in a well populated city with a decent market and people who at least don't hate her (a factor that is often lost for people playing an exotic race like Dragonborn or Thri-kreen). Then we examine the location on a wider scale; assuming you are using a typical forested hilly medieval Europe type background, wood and stone are in no shortage. These two factors tell us that the building will be reasonably cheap to make, maybe no more than 300-800 gold for nails, wood, stone, cement, pipes, and construction services rendered. This is assuming we are only building a modest to respectable home and hearth, so if she wants a mansion add about 1300 GP to either price. This works whether building a new house or if the player is buying an existing property. Renting should probably be cut down to a 10th of the price but, as others have mentioned, deducted from monthly living costs.
Next comes the housing market. The city is a desirable area for it's safety under military protection, conveniences and sheer numbers, so let's say that adds 500 GP for both location and taxes, assuming the local leader isn't a tax lobbing blood sucker. Assuming the area is dangerous outside of town for all the nasty beasties, living just outside the city gates could so much as halve the overall cost depending on what the land offers. Land that could potentially be usable farmland or mined for valuable ore is obviously more valuable than barren soil with no real value. If there are any major threats like bandit raids or dragon attacks, the city might even be funded by the kingdom to pay citizens just to live there (like Alaska).
There is also the population vs. homes available question. If there are a lot of people who want to live there a player may have to pay up an extra 100-300 just because everyone wants a house. Vice versa, if a plague recently killed off everyone the city may be desperate and sell houses at low costs just to get a tenant to pay taxes, so give or take 200 gp off in that case. If status quo, (i.e there's enough houses to go around, neither surplus or deficit) then the straight house cost is fine unaltered.
All of that doesn't account for furnishings. As is, the house is an empty building. Unless the player has furniture available from clearing out a building infested with monsters and said "I'm taking the armoir, couch and bed" then that will also be added to the price as either a pre-furnished bonus cost or in the form of her going to "ye olde carpenter's shop". On a side not this could make for an interesting quest addition.
"You're in the dragon's layer, and fighter, at the far side of the
cave you see the most plush, delightful looking couch you have ever
set eyes on. And over to the left there is a darling wardrobe with
frickin' gold leaf coated door carving and sapphire accents on the
Then you get to play stealthily-carry-a-couch-over-a-pile-of-gold-without-waking-the-dragon!
Another thing to consider is who owns the land. If it was awarded to the player for helping the city she might just need to worry about building costs. One of the things in the players handbook encourages giving land deeds as alternative payment for services as an adventurer.
Before delving any further into cool stuff you could also do, lets tally up the score; 650 gold for a medium sized, comfortable home with a guest room, plus 250 gold for living just outside the city's protecting gates after taxes, plus another 557 GP to furnish it with beds, chairs, tables, a couch, a decent rug, lamps, plates and silverware, and a decorative painting of a gold dragon flying into the sunset over the hearth/fireplace.
The grand total comes out to 1,457 GP.
All this of course is for a mundane, on-the-edge-of-town building that just meets the fighter's standards and isn't a shack with a bed or a tavern room. You can do better than THAT, can't you? I personally have a book in the works of different buildings the players can use as a home or base. Each one has a unique look, theme and floor plan and each has a story. This is where you could have fun sketching out floor plans and drawing buildings. The first I made that started this was a lich's lair dungeon I made; the local paladin chapter had killed the lich, but the players had to go through and mop up all of the undead (players weren't going for my main hook so I sent them there from a "help wanted" board). They cleared through this small mansion observatory pretty quickly, but I loved my original floor plans and concept for the building so much I wanted to keep it around. So I re-did the design a little and Chateau Necro is now one of 5 current properties I have available for purchase. This is also where I thought of "clear to claim"
What does a king do with the illegally built fortress that the bandits fight off all attempts to be evicted from? It would be a lot of time and work to demolish it, and despite it's crude design it does class up the roads leading into town. So what if the building would go to any adventurer who cleared the ruffians out? King gets an easy solution to his problem of the structure AND knows where to reach our heroes in the event of another problem rising as well as new tax payers. The players have a place to shack up and store all their sick loot while they plan to murder the king and usurp his throne. Win-win!
A decent house that fulfills the needs presented would be under 2,500 GP, but there's a lot you could also do with this interest, ranging from quests to motivation for the character in question.