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 4,000,000). 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 is still the same up to many significant digits. So you still can trace back the choice of the Avogadro constant to choices made when defining the meter, the kilogram and the mole.