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I understand that the mole is an extremely useful way of measuring and understanding atomic quantities. I did find some history of the number in its wikipedia article, but I did not find an answer to the question "Why is the number chosen to be what it is?".
I understand that it is not necessary to know the history of the number in order to apply it, but it would seem to me that the number wasn't chosen arbitrarily. Why is it, then, that Avogadro's number was chosen as $6.022\times10^{23}$?
Why was Avogadro's number chosen to be the value that it is?
Your question implies that you already know that it was a choice rather than something derived from first principle. There are some numbers that are derived from first principles. In math, $\pi$ and $e$ are not a choice, but can be derived from their properties. In physics, the fine structure constant is not a choice, but would be measured to get the same value by aliens on a different planet (at least as far as we can tell).
[...] Avogadro's number [...]
To be correct, it is actually Avogadro's constant $N_A$ that we use more often in chemistry (for identities such as $k_B = R / N_A$, where $k_B$ is the Boltzmann constant and R the universal gas constant). Avogadro's constant has the dimension (in the SI unit system) of one divided by amount of substance. So the number in front of the unit will change as you change the unit, just like 30 minutes is the same as 0.5 hours even though 30 is not the same as 0.5.
I also understand that it is not necessary to know the history of the number in order to apply it. But it would seem to me that the number wasn't chosen arbitrarily.
The value of Avogadro's constant is linked to the choices that were made when setting up the metric system. There is one definition, that of the meter, that is linked to our planet (so aliens on another planet would have no reason to choose it). The meter was chosen to be approximately 1/10,000,000th of the quadrant of the Paris meridian (i.e distance from Paris to South Pole to North Pole to Paris divided by 40,000,000). Then, they made an artifact to define the meter exactly. Next, the kilogram was chosen to be approximately the mass of 1/1000th of a cubic meter of water (at some temperature). It is no coincidence that the density of water is very close to 1 kg/L. Again, they made an artifact to define the kilogram exactly.
Once the kilogram was defined, the mole was chosen to contain the same number of particles as (pick your favorite, they are all approximately correct at this point) one gram of protons or neutrons, 16 g of oxygen atoms or 12 g of pure carbon-12 atoms.
How are the definition of kilogram and mole related to Avogadro's constant?
Avogadro's constant tells us how many particles there are per mole of particles. The mole was defined with reference to the gram and fundamental constants such as the mass of a proton or of a single carbon-12 atom, so once these choices are in place, Avogadro's constant is no longer a choice.
Relationship to current definitions of units
The SI units (including meter, kilogram and mole) are defined in a different way today, but their values are still the same up to many significant digits. So you can still trace back the choice of the Avogadro constant to choices made when defining the meter, the kilogram and the mole.
From the practical historical perspective chemists and physicists needed a number to use as a conversion factor from Daltons to grams to perform stoichiometric reactions.
In the beginning, Dalton proposed to use the mass of the Hydrogen and assign it to a value of 1 Dalton without knowing how many grams correspond to 1 Dalton. Scales use grams, not daltons so scientist needs a conversion factor. To do that we could take a gram of hydrogen and count how many atoms are there this would return what is called Avogadro constant. In "theory" this will give the same result also if we take 16 grams of Oxygen (that has 8 protons and 8 neutrons so roughly 16 Daltons) and the results should be the same (of course in practice we would have to take 32 grams of molecular oxygen $\ce{O2}$ because atomic oxygen is not stable. In practice is not like this because there are other factors related to the fact that as Einstein noted mass is also related to energy. Physicists however used Oxygen-16 for a while, then they team up with IUPAC and decided to use 12 grams of Carbon-12. The values of Avogadro costant changed a bit but they finally now after the 2019 redefinition of the SI base units it has been fixed to a value. This is a table form this article (Jensen, William B. "Why Has the Value of Avogadro’s Constant Changed over Time?." Journal of chemical education 87.12 (2010): 1302-1302).
Table 1. Example Experimental Values for Avogadro's Constant over Time*
|------+-----------+-----------|
| Year | Author | Nᴬ/10^23 |
|------+-----------+-----------|
| 1908 | Perrin | 6.7 |
| 1917 | Mullikan | 6.064 |
| 1929 | Birge | 6.0644 |
| 1931 | Bearden | 6.019 |
| 1945 | Birge | 6.02338 |
| 1951 | DuMond | 6.02544 |
| 1965 | Bearden | 6.022088 |
| 1973 | Cohen | 6.022045 |
| 1987 | Deslattes | 6.022134 |
| 1994 | Basile | 6.0221379 |
| 2001 | De Bièvre | 6.021339 |
* A more complete list can be found in ref (3).
No, in fact, the choice of the element was based on some practical considerations initially was preferred oxygen because the way they measured the weight was using ratios between the compounds citing IUPAC publication:
Berzelius published a series of articles on atomic weight measurements in which he reported actual weighings along with results from other chemists on an oxygen scale i.e., value of O = 100. Berzelius felt that oxygen was a good reference material because it combined with most elements (in inorganic chemistry) and many molecular and atomic weights could be measured directly. The drawback to hydrogen as a reference was that it combined directly with a limited number of elements only and the atomic weight must be evaluated in comparison with oxygen. As the oxygen to hydrogen ratio could never be determined with absolute accuracy, any errors in this ratio would be carried over into all other values of atomic weights.
When finally they start using mass spectrometry for determining weights they switched to carbon:
In April 1957 at the bar in the Hotel Krasnapolski in Amsterdam, Nier suggested to Mattauch that the 12C = 12 mass scale be adopted because of carbon's use as a secondary standard in mass spectrometry63. Also, 12C = 12 implied acceptable relative changes in the atomic weight scale, i.e., 42 parts-per-million (ppm) compared to 275 ppm for the 16O = 16 scale (which would not acceptable to chemists). Enthusiastically, Mattauch made a worldwide effort in the late 1950s to publicize the 12C = 12 scale and obtain the physicist's approval, while Wichers obtained the chemist's approval.
Initially the Avogadro was defined as the number of atoms contained in 1 g Hydrogen. Later on, it was understood that Hydrogen can contain various amounts of Deuterium. So that, instead of referring to the fluctuating H atom, it was decided that the Avogadro would be related to the weight of another atom. There had been long discussions about the choice of this atom. Oxygen-16 and Carbon-12 were proposed. After discussion, the atom that is 12 times heavier than the H atom was chosen, namely the Carbon-12 atom. So the Avogadro number was up to recently, the number of Carbon-12 which is contained in $12.000$ g $\ce{C-12}$. But one or two years ago, it was decided that the Avogadro will not be related to any atom, and that it is a given number, which cannot be changed later on if some not yet known technical difficulty would happen in the future.
Other answers have covered it adequately. But there's one other way in which Avogadro's constant is not completely arbitrary. It's an order-of-magnitude measure of the quantity of particles that can sustain sentience on an Earth-type planet. 12 grams of carbon-12 is a sample of matter that's roughly on the same scale as us humans.
I'm serious here. How many atoms does it take to build a self replicating molecule? How many molecules per a cell that's capable of forming multicellular organisms? How many cells per a sentient lifeform? The answers to those questions clearly have a lower bound. Avogadro's constant is the measure of minimal required complexity of sentience. We don't have a lot of reference points on all three - DNA/RNA (citation needed), eucaryotic cell, human. That said, if there was a simpler way to build a cell, or a cheaper way to sustain a central nervous system of 15 billion neurons, evolution would've probably already stumbled upon it, what with the humongous amount of thermal noise it had to work with.
In that way, those aliens on a different planet from Karsten Theis' answer, were they to come up with a concept similar to mole, might have the their coefficient of proportionality roughly in the same ballpark.
See also: Kolmogorov complexity.
