How digital currency (like Bitcoin) proponents solve the problem of its high volatility to make it a good candidate to replace fiat? I have recently read an article about a family living with no fiat currency for several years. Indeed there are lots of advantages of using a digital currency.
I am wondering if having an important fraction of transactions relying on a digital currency is possible (basically replacing fiat with digital currency for an important part of the economy). The first counter-argument for this being possible is the high volatility of the digital currencies.
I have very little economical knowledge, but almost everything from small businesses to the entire country economy seems to rely on currency stability (e.g. inflation rate should be small enough and EU actually forces a maximum value, business plans require relative currency stability, people expect a rather uniform or at least predictable monthly income).
How digital currency proponents solve the problem of its high volatility to make it a good candidate to replace fiat?
tl;dr:
fiat money can be fully digital money as noted in the +1 comment of Jasper so it does not make much sense to talk about replacing fiat money with digital money as fiat money can be fully digital - those are two separate categories.
But I understand that your question is  about bitcoin/cryptocurrency not just any digital money. There are several proposals on how to make bitcoin competitive (as there are not many economists who think it could replace fiat money). These proposals address the biggest weakness of the cryptocurrencies which is inflexible money supply and range from pegging cryptocurrency value to some other currency to inflation targeting.
Full Answer:
Part 1: Clearing out Misconceptions
I think this question is based on confusion about what fiat money and digital money is so I will clear that misconception first. By definition, following Mankiw Principles of Economics pp 606:

fiat money:
money without intrinsic
value that is used as
money by government
decree

Fiat money can be both physical (for example paper notes, coins etc.) and can also be digital (money on you debit card). Also note, not all physical paper notes would be fiat money. Under gold standard where paper notes are backed by gold there would no longer be any fiat money because then the value of money is not created by government decree anymore. So whether money is fiat or not has no relationship to what medium is used as money (paper/bits etc.).
If you want a comparable term that distinguishes bitcoin from fiat digital currency, the term would be &quot;fiduciary money&quot;. Fiduciary money is basically a 'version' of fiat money where the value is not created by government decree but solely by trust (again fiduciary money can be both in form of physical paper and digital). In fact this is also how economists categorize bitcoin. According to Velde (2013):

Bitcoin is a fiduciary currency
Fiduciary currencies—in contrast with
commodity-based currencies (such as
gold coins or bank notes redeemable in
gold)—have no intrinsic value, and derive their value in exchange either from
government fiat or from the belief that they may be accepted by someone else.
They are inherently fragile; government
orders can be ignored or doubted, and
a currency that has value only because
of the belief that it will have value may
have no value at all (for instance, if I
believe that no one will accept it, I will
not accept it either).
The term “mining” may lead one to
think that bitcoin is not fiduciary. It
is true that real resources (computing
hardware and energy) are expended
in creating bitcoins. Since entry to mining is free, the value of these resources
will be the market value of bitcoins produced (whose number per day does
not depend on the size of the network).
But once created, the bitcoin has no
value other than in exchange, contrary
to a gold coin.

Part 2: How to Solve the Problem of Volatility
First, it is important to understand little bit about what determines the price level in terms of some money, where 'value' of money is inversely related to the price level. For example, the US-bitcoin exchange rate ($S$) should be given by $S=\frac{P_{US}}{P_{Bc}}$, where $P_{US}$ is price level (aggregate prices) in terms of USD and $P_{Bc}$ in terms of bitcoin. This needs to hold otherwise if the equality would be violated one could just buy bundle of goods in bitcoin sell it for US dollars and then exchange dollars back to bitcoin for riskless profit (or vice versa depending on how the equality is violated). If there would exist such opportunity it would quickly disappear because as people would take advantage of the exchange the demand for dollars or bitcoin would change until it would reach equilibrium where the above would hold. So there is an inverse relationship between value of bitcoin and its price level.
Broadly speaking the price level for any money is determined by money market equilibrium.  In its simplest form this can be described by equation of exchange (See Mankiw Macroeconomics pp 87) as:
$$MV=PY$$
Where $M$ is the money supply, $V$ velocity of money, $P$ price level and $Y$ output (this is oversimplification, expectations of these quantities matter as well but since you are not an economist I wanted to only use most basic models to provide structure). Solving for price level and log-linearizing we get:
$$\ln P = \ln M + \ln V - \ln Y $$
So if money supply would increase by $1\%$ you would expect price level (decrease in value) to increase by $1\%$, holding all else equal, the same for velocity and for real output you would expect decrease in price level (of course in real life there might not be 1:1 proportionality but again this is just to provide some structure for our thinking - in addition with bitcoin which is in some sense close to commodity money the production costs of mining new bitcoin would play a role as well, but lets assume that away for now for simplicity).
Here you can nicely see what the problem with cryptocurrencies vis-à-vis standard fiat money is. Most of cryptocurrencies  have a very inflexible money supply. For example, money supply of bitcoin cannot normally decrease - there is no way how you can 'un-mine' properly mined bitcoins, and there is hard cap on what maximum number of bitcoins ever will be. This means that for bitcoin $M$ will eventually become fixed. This means that its value (inverse of the price level) will be always at mercy to the shocks to velocity or real output (and in more complex models expectations of those). In fact according to the Iwamura et al (2019), the inflexibility of money supply is precisely the main reason what causes the price instability of crypto currencies.
Fiat money monetary system does not have such problem as central bank can, at least in principle, always choose what sort of $M$ it wants, and those central banks that have mandate for stable prices try to do so in a way that keeps prices stable (although usually this is done in indirect way through interest rates as opposed to directly choosing some $M$ - but in principle they could choose any $M$ if they  would want to).
Now finally we can turn to your main question:

How digital currency proponents solve the problem of its high volatility to make it a good candidate to replace fiat?

Well coincidentally there is whole paper that tries to address this question, namely:
Iwamura,  Kitamura, Matsumoto,  &amp; Saito (2019). Can we stabilize the price of a cryptocurrency?: Understanding the design of Bitcoin and its potential to compete with Central Bank money
They basically argue that in order to make bitcoin or other cryptocurrencies competitive there must be some way how to make supply of money flexible both upward and downward. Their potential fixes include:

Currency Peg/Currency rule: The cryptocurrency could be pegged to some  real life currency, where there would be some rule that would adjust money supply in a way to always keep exchange rate between lets say USD and Bitcoin or euro and Bitcoin fixed. Alternatively, there could be some less restrictive rule where bitcoin would have to follow other currency within some band.

Monetary Policy without a Central Bank:  you can simply have some monetary rule (some argue central banks follow monetary rules such as the Taylor rule) that affects money supply by making it easier or harder to mine bitcoins that depends on some parameters in economy (for example, real output).

Inflation Target in Cryptocurrency: crypto currency money supply could be set up in a way that it always targets lets say $2\%$ inflation (or zero inflation - which would be perfect price stability, but most economists argue small positive inflation is vital for economy (see more on that here).


However, the above being said I do not know of any conventional economist who would think that cryptocurrencies should replace fiat money. Most macroeconomist believe that state should manage business cycle using, among other tools, monetary policy. In addition to that cryptocurrencies have further problems as mentioned in the paper cited above:

First, cryptocurrencies are more expensive to produce, and the production costs are hard to retrieve. Bank notes issued by the central banks require some printing and material costs. These costs are negligible compared with the face (nominal) value.
Second, bank notes are reversible between new issues and absorption because the central bank basically buys and sells securities with bank notes. A cryptocurrency cannot be absorbed, but if equipped with a built‐in value stabilization mechanism, this shortfall of irreversibility can be softened (but not eliminated) in practice.
Third, Bitcoin‐type cryptocurrencies use a delayed finality confirmation structure to avoid double spending. Consequently it typically takes hours to use obtained money. Bank notes can be used  immediately as obtained. Fourth, Bitcoin type cryptocurrencies face security risks, such as Denial of Service attacks, more widely than bank notes.

Some of the above can be aleviated further, as authors explain:

The third and fourth points are relative problems, and also intrinsic to Bitcoin‐type currencies. They are not general problems with cryptocurrency. The third point considers a problem directly compared with bank notes for direct transactions. Considering transactions with Bitcoin‐type currency may occur over a remote distance, finality confirmation may be quicker and much cheaper with a Bitcoin‐type currency than that through a bank. The fourth point is closely related to the protocol design of a Bitcoin‐type currency, and is not a general cryptocurrency issue. The instability associated with mining pools due to strategic behaviors between and within pools can be reduced substantially if the valuation system is improved in line with our suggestions.

However, the first and second point are inherent problems of cryptocurrencies, as authors further explain:

The first and second points are fundamental shortfalls of cryptocurrency. As currently described, cryptocurrency values are based on associated production costs. This mechanism is similar to commodity money, notably gold and silver coins. Historically gold and silver coins have been replaced by credit (or fiat) money basically because of the above‐mentioned first and second points.

Consequently, at best it is reasonable to aim to make crypto currencies competitive vis-à-vis fiat currency, rather than as their replacement.