Why does the metric system use "kilogram" as a base SI unit?

Among the base units of the International System, the kilogram is the only one whose name and symbol, for historical reasons, include a prefix. Names and symbols for decimal multiples and submultiples of the unit of mass are formed by attaching prefix names to the unit name "gram", and prefix symbols to the unit symbol "g" (CIPM 1967, Recommendation 2).

^BIPM

The reason why "kilogram" is the name of a base unit of the SI is an artefact of history.

Louis XVI charged a group of savants to develop a new system of measurement. Their work laid the foundation for the "decimal metric system", which has evolved into the modern SI. The original idea of the king's commission (which included such notables as Lavoisier) was to create a unit of mass that would be known as the "grave". By definition it would be the mass of a litre of water at the ice point (i.e. essentially 1 kg). The definition was to be embodied in an artefact mass standard.

After the Revolution, the new Republican government took over the idea of the metric system but made some significant changes. For example, since many mass measurements of the time concerned masses much smaller than the kilogram, they decided that the unit of mass should be the "gramme". However, since a one-gramme standard would have been difficult to use as well as to establish, they also decided that the new definition should be embodied in a one-kilogramme artefact. This artefact became known as the "kilogram of the archives". By 1875 the unit of mass had been redefined as the "kilogram", embodied by a new artefact whose mass was essentially the same as the kilogram of the archives.

The decision of the Republican government may have been politically motivated; after all, these were the same people who condemned Lavoisier to the guillotine. In any case, we are now stuck with the infelicity of a base unit whose name has a "prefix".

^BIPM

The International Bureau of Weights and Measures (French: Bureau international des poids et mesures), is an international standards organisation, one of three such organisations established to maintain the International System of Units (SI) under the terms of the Metre Convention (Convention du Mètre). The organisation is usually referred to by its French initialism, BIPM.

^Wikipedia

The previous base unit, gram, was replaced by the kilogram in order to obtain coherence with the practical units ampere and volt.

In 1874 the mechanical units cm, g, s ('CGS') were adopted as the coherent system of units for science. In 1881 the CGS was coherently extended with the “absolute" electrical units abampere, abvolt, abohm. Coherence in this case primarily means that electrical energy and mechanical energy have identical units: $V\ I\ t = F \ L$. Unfortunately, the abvolt, abohm, were inconveniently small. Another inconveniently small unit was the unit of mechanical energy, the erg (=1 g⋅cm/s2).

In 1881-1889 the 'practical' units electrical units ampere, volt, ohm, and joule (1 joule = 1 V⋅A⋅s = 10^7 erg) were introduced for practical use. Their magnitude was convenient, but they were not coherent with CGS. In 1901, Giorgi pointed out that the practical units are coherent with m, kg, s.

Derivation: $[E] = [F\cdot L] = [(M\ L\ t^{-2})\ L] = [M]\ [L]^{2}\ [t]^{-2}\rightarrow$ $\rightarrow [M] = [E]\ [t]^2\ [L]^{-2} = 1\ J s^{2} m^{-2} = 1\ kilogram$

Because of this, the scientific community adopted the SI system in 1960, with the kilogram as base unit.

In summary: the kilogram became the base unit because it is coherent with the joule, which is derived from the practical units volt and ampere.

(source: Jayson, Amer. J. Phys. 82 (2014) 60)