An early notation to Christoffel symbols is:
\begin{equation} \begin{Bmatrix} m \\ \hspace{1.5mm}b\hspace{5mm} c\hspace{1.5mm} \end{Bmatrix} = \frac{1}{2}g^{ma}\{ \partial_{b}g_{ca} + \partial_{c}g_{ab} - \partial_{a}g_{bc}\} \tag{1} \end{equation}
with these curly brackets.
Now, in Einstein's notebook and in the seminal paper of Einstein Field Equations we see a clear usage of this notation:
and
I'm not sure if was Einstein indeed the first one to use this curly notation (or the german publishers). So, It was Elwin Christoffel the one who introduced this notation? If no, would like to know the one who first have used this notation.
The bracket notation and brace notation was the first notation, due to Christoffel and appearing in his 1869 paper. See both the 3rd and 4th pages.
The $\Gamma$-notation is due to Einstein in his work on general relativity.
This is discussed, with links to a bunch of papers by different authors (Christoffel, Einstein, Ricci, Weyl, et al.), in the question I asked about this issue a couple of years ago here. It is suggested in comments there that the use of $\Gamma$ by Einstein might be an allusion to "gravitation" and the $\Gamma$-notation should be called Einstein symbols instead of Christoffel symbols.