Would House ever have any advantage to gain by controlling the dominant currency? House coins, rather than caps or NCR dollars? Or is it completely irrelevant to his power?

Currency has to be backed up with value.
In Fallout the currency that has the biggest value is caps (as much as i know fallout lore).
If House wants to create something like "House coins" maybe they will have the value of caps in the Mojave becase of House power over New Vegas and its wealth.
But i think that outside of the Mojave it will not be the same story.
The NCR and Legion bouth have ther own currency that have value backed up by gold and stability.
The NCR value of NCR dollars is less than caps because of their war with the Brotherhood

Spoiler:

 

The gold in Redding, for a while, backed the NCR currency. However, by 2281, the gold had been targeted and destroyed during the NCR-Brotherhood War. As a result, the currency has become severely devalued, with a hundred-dollar bill coming out to around 40 caps rather than its face value

and the Legion coins have bigger vallue

Spoiler:

 

Legion currency is still accepted as payment in the Mojave Wasteland, owing to the high level of trade with the Legion and the safety of its territory for caravans. The exchange rate is 4 bottle caps to 1 Denarius, and 100 bottle caps to 1 Aureus.

becase of the safety for the caravans that work whit them.
If House wanted to create a currency and make the Mojave accept it he would have to back it up with gold and power for long time in order for it to take on the market and be accepted and then win at the Hoover Dam in order for the value to be the same outside of the Mojave.
The information for the "Spoilers" was taken from Fallout Wiki.
Link for NCR dollar --> http://fallout.wikia.com/wiki/NCR_dollars
Link for Legion Denarius --> http://fallout.wikia.com/wiki/Legion_Denarius

I cannot really say if he would have advantage of making his own currency in this Post-Apocalyptic world.
I think that the only real currency in the game are caps and the rest is a show-of for power (which alone is advantage (so maybe yes)).

p.s. Thank you for the topic.

Money should be backed by value, but... ah Bethesda.  With bottlecaps inexplicably being used everywhere, it's difficult to understand.  Obsidian and old-school Fallout at least seemed to understand that value comes from backers, which can rise and fall.
The Hub water merchants with their Bottlecaps and the brotherhood both felt that there was a reason to compete with the NCR dollars.  So maybe House would too, especially after a win over the NCR in NV ... Lucky 38 chips? I can imagine casino chips being worth money eventually.

Edit: While I'm here, I might as well ponder whether or not the Enclave would/could/should have tried to re-introduce the value of pre-war money. It's not completely worthless as it is, can still be sold to merchants.

Well the casino chips have value but i think that it is only recognised in Vegas.
For the Enclave they really didnt had to be seen as a power in the wasteland because of their plans to kill all mutated life which includes the people that use the currency (which is basically all of the post-apocalyptic America) so i think that they would had have to make or use old currency after the success of the "big cleanup" which never happened in its full.

What would it mean if a country is using another leader's coins rather than their own? The NCR dollars are failing and probably dying out as of the start of New Vegas -- that has to have actual meaning at least by an understanding of a real-world economy.

@Sirdanest wrote:

What would it mean if a country is using another leader's coins rather than their own?  The NCR dollars are failing and probably dying out as of the start of New Vegas -- that has to have actual meaning at least by an understanding of a real-world economy.

So, in a bartering system like Fallout, the benefit of currency is primarily a question of convenience vs liquidity; coins and paper money take up a lot less space and weigh much less than trade goods, but are only worth anything if you can transform them back into stuff at the same rate, or better.

A preference for any currency there would be the same as here; it means the money is more accepted, easier to change into goods, more accessable; etc. Materially, this chains into a lot of little things that add up; Legion denarii being accepted everywhere improves relations with local communities, makes it easier for merchants to talk up their goods, etc.

The really interesting problems arrive from minting and counterfeiting; historically, the best way to make your money worth something was to collect it for taxes. This also let you control its creation, which means you're not broke because you can just make more money. Using another nation's currency breaks that part, forcing a government to either try to print the other nation's currency, or lose the option to cheat at finances. Materially, this all undermines faith in the weaker nation and boosts up faith in the stronger one, which takes shape in a million little ways.

So... it's not good to have to use someone else's money. It's hysterically bad to have to use your enemy's.