The Gini coefficient is commonly used as a measure of inequality of income, consumption or wealth. There was an attempt to apply the Gini index to measure inequality in the HDI distribution (Hicks, 1998).
The choice of the Atkinson inequality measure was guided by three factors: (i) subgroup consistency, (ii) sensitivity to the inequality in the lower end of distribution, and (iii) simplicity of computation and mathematical elegance of the resulting composite Inequality-adjusted Human Development Index.

Subgroup consistency means that if inequality declines in one subgroup (region, ethnic group, etc.) and remains unchanged in the rest of population, then the overall inequality declines. The Gini coefficient does not have this property.
By its construction, the Gini coefficient puts equal weights to the entire distribution, while the Atkinson inequality measure puts more weight to the lower end, thus it accounts better for child mortality, illiteracy, and income poverty.
Finally, the geometric form of the HDI in combination with the Atkinson index provides a simple and elegant composite IHDI, obtained by first computing inequality for each dimension and then across dimensions, which further implies that it can be computed by combining data from different sources (life tables and different surveys for education and income).