Como rotular um ponto no eixo?

Question

Use yshiftpara mudar o rótulo. O nó com o rótulo se torna:

\node[label={[yshift=2ex]180:{($\varepsilon_{cor}$)}},circle,fill,inner sep=2pt] at (axis cs:0.5066,-5.4) {};

Deixando o resto como está agora, o resultado é:

EDITAR

Como foi solicitado pelo OP, incluo outra solução. Para posicionar o rótulo à direita do visto você deve especificar label={[right,yshift....

Tomei a liberdade de alinhá-lo manualmente

\documentclass[tikz]{standalone}
\usepackage{pgfplots}
\usepackage{amsmath,amssymb}
\usepackage{siunitx}
\begin{document}
\begin{tikzpicture} 
    \pgfplotsset{width=10cm,}
        \begin{axis}[
            xlabel={$\varepsilon$ /\si{\volt}},
            ylabel={$\lg \vert I \vert$},
            axis lines = box,
            xmin=0.18, xmax=0.82,
            ymin=-5.4, ymax=-2.9,
            xtick={0.2,0.3,0.4,0.5,0.6,0.7,0.8},
            ytick={-5,-4.5,-4,-3.5,-3,-2.5,-2},
            ymajorgrids=true,
            grid style=dashed,]
        \addplot[
            color=black,
            mark=*, mark size=1.5,
            ]
            coordinates {
            (0.2208 ,-3.05061)
            (0.2809 ,-3.1549 )
            (0.3309 ,-3.26761)
            (0.3709 ,-3.38722)
            (0.4008 ,-3.49485)
            (0.4208 ,-3.58503)
            (0.4309 ,-3.63827)
            (0.4408 ,-3.69897)
            (0.4508 ,-3.76955)
            (0.4609 ,-3.85387)
            (0.4708 ,-3.95861)
            (0.4759 ,-4.04576)
            (0.4808 ,-4.09691)
            (0.4858 ,-4.22185)
            (0.4909 ,-4.30103)
            (0.4958 ,-4.52288)
            (0.4978 ,-4.52288)
            (0.4998 ,-4.69897)
            (0.5019 ,-5      )
            (0.5039 ,-5      )
            (0.5063 ,-5.41   )}; 
        \addplot[
            color=black,
            mark=*, mark size=1.5,
            ]
            coordinates {
            (0.5066 ,-5.41   )  
            (0.5101 ,-5      )  
            (0.5121 ,-5      )  
            (0.5141 ,-4.69897)  
            (0.5161 ,-4.69897)  
            (0.521  ,-4.39794)  
            (0.526  ,-4.30103)  
            (0.5311 ,-4.1549 )  
            (0.536  ,-4.09691)  
            (0.5411 ,-4      )  
            (0.5511 ,-3.88606)  
            (0.5611 ,-3.79588)  
            (0.5711 ,-3.72125)  
            (0.5811 ,-3.67778)  
            (0.5911 ,-3.61979)  
            (0.611  ,-3.52288)  
            (0.6412 ,-3.40894)  
            (0.6811 ,-3.30103)  
            (0.7311 ,-3.19382)  
            (0.7911 ,-3.09691)}; 
            \node[label={[right,yshift=2.1ex,xshift=0.3em]180:{($\varepsilon_{\mathrm{cor}})=0.5066$}},circle,fill,inner sep=2pt] at (axis cs:0.5066,-5.4) {};
        \end{axis}
    \end{tikzpicture}
   \end{document}

Produzindo:

EDITAR2

Reposicionei a etiqueta no eixo.

A parte principal é:

extra x ticks={0.5066},
        extra x tick labels={$\varepsilon_{\mathrm{cor}} ({\scriptstyle 0.5066})$},

Além disso, eu meio que corrigi a marca do eixo:

        \node[circle,fill,inner sep=1.5pt] at (axis cs:0.5066,-5.4) {};

O código é:

\documentclass[tikz]{standalone}
\usepackage{pgfplots}
\usepackage{amsmath,amssymb}
\usepackage{siunitx}
\begin{document}
\begin{tikzpicture} 
    \pgfplotsset{width=10cm,}
        \begin{axis}[
            xlabel={$\varepsilon$ /\si{\volt}},
            ylabel={$\lg \vert I \vert$},
            axis lines = box,
            xmin=0.18, xmax=0.82,
            ymin=-5.4, ymax=-2.9,
            xtick={0.2,0.4,0.6,0.8},
            ytick={-5,-4,-3,-2},
            ymajorgrids=true,
            extra x ticks={0.5066},
            extra x tick labels={$\varepsilon_{\mathrm{cor}} ({\scriptstyle 0.5066})$},
            grid style=dashed,]
        \addplot[
            color=black,
            mark=*, mark size=1.5,
            ]
            coordinates {
            (0.2208 ,-3.05061)
            (0.2809 ,-3.1549 )
            (0.3309 ,-3.26761)
            (0.3709 ,-3.38722)
            (0.4008 ,-3.49485)
            (0.4208 ,-3.58503)
            (0.4309 ,-3.63827)
            (0.4408 ,-3.69897)
            (0.4508 ,-3.76955)
            (0.4609 ,-3.85387)
            (0.4708 ,-3.95861)
            (0.4759 ,-4.04576)
            (0.4808 ,-4.09691)
            (0.4858 ,-4.22185)
            (0.4909 ,-4.30103)
            (0.4958 ,-4.52288)
            (0.4978 ,-4.52288)
            (0.4998 ,-4.69897)
            (0.5019 ,-5      )
            (0.5039 ,-5      )
            (0.5063 ,-5.41   )}; 
        \addplot[
            color=black,
            mark=*, mark size=1.5,
            ]
            coordinates {
            (0.5066 ,-5.41   )  
            (0.5101 ,-5      )  
            (0.5121 ,-5      )  
            (0.5141 ,-4.69897)  
            (0.5161 ,-4.69897)  
            (0.521  ,-4.39794)  
            (0.526  ,-4.30103)  
            (0.5311 ,-4.1549 )  
            (0.536  ,-4.09691)  
            (0.5411 ,-4      )  
            (0.5511 ,-3.88606)  
            (0.5611 ,-3.79588)  
            (0.5711 ,-3.72125)  
            (0.5811 ,-3.67778)  
            (0.5911 ,-3.61979)  
            (0.611  ,-3.52288)  
            (0.6412 ,-3.40894)  
            (0.6811 ,-3.30103)  
            (0.7311 ,-3.19382)  
            (0.7911 ,-3.09691)}; 
                       \node[circle,fill,inner sep=1.5pt] at (axis cs:0.5066,-5.4) {};
        \end{axis}
    \end{tikzpicture}\end{document}

E aqui está o resultado

Basicamente:

Reduzi os ticks em ambos os eixos (você pode fazer como quiser)
Forneci uma marca extra no eixo x, comextra x ticks
Forneci um rótulo para a marca extra (você pode querer ajustar o estilo, por exemplo, com um \smallou revisar os números)
Retirei a etiqueta da trama, sendo redundante
Reduzi inner sepo tick único no eixo para 1.5pte pode exigir ajuste fino

Acho que você pode pegar aqui.

Answer 1