What are the reasons for making the mole a base SI unit? We have meter in the SI - we use that to measure length. Other length units like light years can be expressed in meters. But how often do we express amounts or quantities in moles? Mole is a number of things, nut we do not define 1 in terms of mole, or a dozen, or a million. I have never seen anyone stating that a coil has $500/N_A$ moles of turns. It seems that we define none of the everyday counting units in moles.
However, it is included in SI, and it is even proposed to be included in the revisions. Why not just use 1 if we need the reference amount for the system of counting units, and make mole a derived or a non-SI unit used in SI (like liter)?
Inspired by Why is the mol a fundamental physical quantity?
 A: Because SI is a system of standards to ensure that different experimental results are reported in compatible formats, not a system of fundamental physical units.  Many of the SI "base units" clearly have no fundamental physical significance - for example, for all intents and purposes temperature and energy conceptually have the same units, and the candela is a unit of "luminous intensity," which is specifically calibrated to the human eye, not to any natural physical quantity.
My advice: if you're doing experiments, stick to SI.  If you doing theory (including classroom-type learning), stick to any other system.
A: How often do we get to use candela, another base SI unit? However, people dealing with intensity of lamps needed it, so it was added:

"Prior to 1948, various standards for luminous intensity were in use in a number of countries. These were typically based on the brightness of the flame from a "standard candle" of defined composition, or the brightness of an incandescent filament of specific design... Germany, Austria and Scandinavia used the Hefnerkerze, a unit based on the output of a Hefner lamp.It became clear that a better-defined unit was needed".

For other stories concerning adoption and reclassification of SI units see Are units of angle really dimensionless?
With mole the reference point is chemistry, and some analogs of mole were used long before it was adopted by SI. In 1865 Loschmidt first estimated the number of molecules in a cubic centimetre of a gas under normal conditions as 1.83 × 1018, and in 1889 determined the gram-molecular volume of gases under normal conditions, after Horstmann introduced the concept of gram-molecular weight in  1881. The term "mole" was introduced in 1900 by Ostwald, a leading chemist at the time, in his textbook. He originally defined it as "the molecular weight of a substance in mass grams", but later clarified "that amount of any gas that occupies a volume of 22414 mL in normal conditions is called one mole". SI only adopted the mole in 1971. 
Johansson is perhaps the most vocal recent critic of the mole, and he also advocates exchanging it for unit one, see his Metrological thinking needs the notions of parametric quantities, units and dimensions:

"The claim of this paper is that metrology would profit from distinguishing between true and parametric quantities, units and dimensions. In particular, these distinctions have repercussions on how to look at the unit one, the mole and the corresponding quantities and dimensions... The third part takes for granted that both the mole and the unit one are parametric units,
  and it argues that, for pedagogical reasons, the mole should be exchanged for the unit one, and the parametric quantity amount of substance be renamed as ‘elementary entities’.

But his reasons are more principled:

"The introduction of the base unit mole in the SI brochure differs in structure from the introductions of all the other six base units of the SI system; it contains two paragraphs, the others only one (corresponding to the first paragraph below). The brochure says:
1. The mole is the amount of substance of a system which contains as many elementary entities as there are atoms in 0.012 kilogram of carbon 12; its symbol is “mol”.
2.When the mole is used, the elementary entities must be specified and may be atoms, molecules, ions, electrons, other particles or specified groups of such particles. [2, p 115]
This means that, strictly speaking, the base quantity at hand is not just amount of substance, but amount (of substance) of elementary entities of a certain kind. This means that, strictly speaking, the base quantity at hand
  is not just amount of substance, but amount (of substance) of elementary entities of a certain kind; briefly, amount(-of substance)-of-$E_p$, where the subscript $p$ functions as a parameter whose “values” are atoms, molecules, ions, electrons, etc. Therefore, the quantity amount of substance had better be called a parametric quantity. The main reason behind the requirement of paragraph 2 is, I take it, that it makes no physical–chemical sense to compare amounts of different kinds of elementary entities.
...to compare an amount of atoms with an amount of molecules would be
  like comparing a number of houses with a number of blocks. When the elementary entities spoken of in the definition become specified, then the mole unit becomes specified. That is, the term ‘mole’ has in practice, ever since its
  introduction in the SI (1971), been used as if it means not just mole but mole-of-$E_p$. The mole is not a base unit on a par with the six property base units; it cannot be used in significant physical–chemical comparisons until the subscript parameter $p$ in mole-of-$E_p$ has been given a certain “value”."

