It seems that
The $J$ invariant is the unique modular function of weight zero for $\operatorname{SL}(2,\mathbb{Z})$ which is holomorphic away from a simple pole at the cusp such that $$J(e^{2\pi i/3})=0,\, J(i)=1.$$
Is the above statement true? If so, can you refer me to a book where this is proved?
To make sure we're on the same page: $$J(\tau)=\frac{E_4(\tau)^3}{E_4(\tau)^3-E_6(\tau)^2}$$ where $E_4$ and $E_6$ are the normalized Eisenstein series, $$E_4(\tau)=1+240\sum_{n=1}^\infty \frac{n^3 e^{2\pi i\tau n}}{1-e^{2\pi i\tau n}},$$ $$E_6(\tau)=1-504\sum_{n=1}^\infty \frac{n^5 e^{2\pi i \tau n}}{1-e^{2\pi i\tau n}}.$$
Let $f$ is a modular function of weight $0$, holomorphic away from the cusp and with a simple pole at the cusp, and $f(e^{2\pi i/3})=0.$ Then $g=f/J$ has no poles and must be constant. But $g(i)=1.$