Banach demonstrated that strong differentiability conditions on a function do not necessarily ensure the almost everywhere (a.e.) convergence of its Fourier series concerning arbitrary orthonormal systems (ONS). Conversely, it is well established that the Menshov-Rademacher theorem provides a sufficient criterion for a.e. convergence of an orthonormal series. We investigate the convergence of the general Fourier series $\Lip\alpha$, when $0<\alpha<1$. In particular, we found special conditions for functions of the orthonormal system, for which the Fourier series of functions of class $\Lip\alpha$ converges. It is established that the resulting conditions are the best possible in a certain sense.