Short answer: the goal of "thermochemical accuracy" for computational chemistry is to match or exceed experimental accuracy. Thus, ~1 kcal/mol comes from the typical error in thermochemical experiments.

The drive began with John Pople, who begin the modern effort to consider "Model Chemistries," comparing the accuracy of different methods across many molecules and often multiple properties. He realized that for thermodynamic properties, one could approach the accuracy of experiments. (See, for example his Nobel lecture).

As the model becomes quantitative, the target should be that data is reproduced and predicted within experimental accuracy. For energies, such as heats of formation or ionization potentials, a global accuracy of 1 kcal/mole would be appropriate.

He then started work on composite methods like G1, G2, G3, etc. that could approach predicting many chemical properties to this accuracy.