documentclass{article}

usepackage{pgf}
 pgfmathsetseed{numberpdfrandomseed}
usepackage{pgffor}
usepackage{tikz}

pgfmathdeclarerandomlist{FracSlopes}
{%
 {frac{1}{2}}
 {frac{3}{2}}
 {frac{5}{2}}
 {frac{7}{2}}
 {frac{1}{3}}
 {frac{2}{3}}
 {frac{4}{3}}
 {frac{5}{3}}
 {frac{7}{3}}
}

pgfmathrandomitem{FracSlope}{FracSlopes}




begin{document}

% Since the line below works
% I know the fraction is being selected randomly.
(y=FracSlope{x})

begin{tikzpicture}[scale=0.3]
draw[help lines, gray, thin] (-10,-10) grid (10,10);
draw[very thick,<->] (-10.3,0)--(10.3,0);
draw[very thick,<->] (0,-10.3)--(0,10.3);
%
% Everythig compiles if the next two lines are commented out.
clip (-10,-10) rectangle (10,10);
 draw[blue, very thick, domain=-10:10] plot (x,FracSlope*x);
 % It's interpreting FracSlope as a string, not a number.
 % I'm *pretty* sure that's the problem.
end{tikzpicture}

end{document}

documentclass{article}

usepackage{tikz}

pgfmathsetseed{numberpdfrandomseed}

pgfmathdeclarerandomlist{FracSlopes}
{%
 {mfrac{1}{2}}
 {mfrac{3}{2}}
 {mfrac{5}{2}}
 {mfrac{7}{2}}
 {mfrac{1}{3}}
 {mfrac{2}{3}}
 {mfrac{4}{3}}
 {mfrac{5}{3}}
 {mfrac{7}{3}}
}

pgfmathrandomitem{FracSlope}{FracSlopes}

newcommand{mfrac}{frac}
newcommand{computefrac}[2]{(#1/#2)}


begin{document}

% Since the line below works
% I know the fraction is being selected randomly.
(y=FracSlope x)

begin{tikzpicture}[scale=0.3]
letmfraccomputefrac
draw[help lines, gray, thin] (-10,-10) grid (10,10);
draw[very thick,<->] (-10.3,0)--(10.3,0);
draw[very thick,<->] (0,-10.3)--(0,10.3);
%
% Everythig compiles if the next two lines are commented out.
clip (-10,-10) rectangle (10,10);
 draw[blue, very thick, domain=-10:10] plot (x,{FracSlope*x});
end{tikzpicture}

end{document}