Usually when transforming into dimensionless variables one looks at the relevant constants in the problem. For hydrogen atom we have the electron charge $e$, electron mass $m_e$, Plank constant $\hbar$, permittivity of free space $\epsilon_0$
Then one does dimensional analysis to make scales based on the above constants. And the expression for length scale turns out to be $$a_0=\frac{4\pi\epsilon_0\hbar^2}{e^2 {m_e}}$$ The extra $4\pi$ is part of the permittivity. And this is exactly the Bohr radius.