I drew a randomly colored grid of points with tikz, how do I force it to remember the first grid from then on?

A simple way is to save the grid to a box:

\documentclass[a4paper,12pt]{article}
\usepackage{tikz}
\usetikzlibrary{calc}

\begin{document}

\newbox{\mygrid}
\savebox{\mygrid}{%
\begin{tikzpicture}[x=0.6cm,y=0.8cm]
\foreach \x in {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18}
\foreach \y in {0,1,2,3}
{
    \pgfmathrandomitem{\RandomColor}{MyRandomColors} 
    \draw[\RandomColor, fill=\RandomColor] (\x,\y) circle (0.2cm);
}
\end{tikzpicture}
 }

%when you want to use the grid, you just call \usebox

\usebox{\mygrid}

\end{document}

You may also place the box within a node for placement control.

Specifying a seed \pgfmathsetseed{1} within your tikzpicture should fix the problem.

\documentclass[10pt,a4paper]{beamer}
\usepackage{tikz}
\usetikzlibrary{calc}

\pgfmathdeclarerandomlist{MyRandomColors}{%
    {red}%
    {blue}%
    {green}%
}

\begin{document}
    \begin{frame}
        \begin{tikzpicture}[x=0.6cm,y=0.8cm]
        \pgfmathsetseed{1}%
        \foreach \x in {0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18}
        \foreach \y in {0,1,2,3}
        {
            \pause
            \pgfmathrandomitem{\RandomColor}{MyRandomColors} 
            \draw[\RandomColor, fill=\RandomColor] (\x,\y) circle (0.2cm);
        }
        \end{tikzpicture}
    \end{frame}

\end{document}

Alternatively to saving the grid in a box, you could compute the random color order only once and use it when creating the grid:

\documentclass{article}

\usepackage{tikz}
\usetikzlibrary{calc}

\usepackage{etoolbox}

\pgfmathdeclarerandomlist{MyRandomColors}{%
    {red}%
    {blue}%
    {green}%
}

\makeatletter
    \def\colorrows{\@gobble}
    \foreach \row in {0,...,3} {
        \def\colors{\@gobble}
        \foreach \col in {0,...,18} {
            \pgfmathrandomitem{\randomcolor}{MyRandomColors}
            \xappto\colors{,\randomcolor}
        }
        \xappto\colorrows{,{\colors}}
    }
    \edef\colorrows{\colorrows}
\makeatother

\begin{document}

\begin{tikzpicture} [x=0.6cm, y=0.8cm]
    \foreach \row [count=\y] in \colorrows {
        \foreach \col [count=\x] in \row {
            \draw [\col, fill=\col] (\x,\y) circle (0.2cm);
        }
    }
\end{tikzpicture}

\end{document}

The \@gobbles get rid of the first comma added by each loop.