PGF/TikZ: How to store strings in array?

To answer the specific question you are asking, here is a complete minimal example.

\documentclass{standalone}
\usepackage{pgfmath}
\begin{document}
\def\names{{"Katie","Frank","Laura","Joe"}}%
\pgfmathparse{\names[2]}\pgfmathresult
\end{document}

So, you need to use the pgfmath parser to do the job through \pgfmathparse and then use the result of the parsing with \pgfmathresult: in other words, you cannot call \names[2] directly.

Here is a more complete example with a \foreach loop:

\documentclass{standalone}
\usepackage{pgfmath,pgffor}
\begin{document}
\def\names{{"Katie","Frank","Laura","Joe"}}%
\foreach \i in {0,...,3} {%
  Name \i: \pgfmathparse{\names[\i]}\pgfmathresult, }
\end{document}

This answer may be more generic than specifically relating to TikZ/PGF.

(La)TeX is a macro-based language, so it does not work as expected compared to other languages when dealing with "arrays". For example, while \names[2] should yield Laura where

\def\names{Katie, Frank, Laura, Joe}

(indexing from 0), (La)TeX considers [2] to have no connection to \names. As such, you're more likely to obtain the output Katie, Frank, Laura, Joe[2] - a concatenation of \names (as it is defined) and [2].

In order to allow for indexing like one typically would using arrays, you would need some other functionality. Here's an example of a list parser that works like you would expect arrays do:

\documentclass{article}
\usepackage{xparse}% http://ctan.org/pkg/xparse
\usepackage{etoolbox}% http://ctan.org/pkg/etoolbox
\newcounter{listtotal}\newcounter{listcntr}%
\NewDocumentCommand{\names}{o}{%
  \setcounter{listtotal}{0}\setcounter{listcntr}{-1}%
  \renewcommand*{\do}[1]{\stepcounter{listtotal}}%
  \expandafter\docsvlist\expandafter{\namesarray}%
  \IfNoValueTF{#1}
    {\namesarray}% \names
    {% \names[<index>]
     \renewcommand*{\do}[1]{\stepcounter{listcntr}\ifnum\value{listcntr}=#1\relax##1\fi}%
     \expandafter\docsvlist\expandafter{\namesarray}}%
}
\begin{document}
\newcommand{\namesarray}{Katie, Frank, Laura, Joe}%
\verb|\names:|\ \names \par
\verb|\names[2]:|\ \names[2] \par
\verb|\names[0]:|\ \names[0] \par
\verb|\names[5]:|\ \names[5]
\end{document}

The idea here is to store the names in an array \namesarray and then define a macro (or "function") that takes an optional argument. If no argument is supplied (i.e., you just use \names), then you print the entire \namesarray. If an argument is supplied (of the form \names[<index>]), parse the list sequentially to find that item that matches <index> and print it.

The list parser relies on etoolbox's \docsvlist and enumerator \do.

The arrayjob package has some more macros for using the data array.

\documentclass{article}
\usepackage{arrayjob}

\newarray\names
\readarray{names}{Katie&Frank&Laura&Joe}

\begin{document}
\verb|\names(3):|\ \names(3) \par
\verb|\names(1):|\ \names(1) \par
\verb|\names(5):|\ \names(5)
\end{document}

Note:
The arrays are 1 based (index starts at 1)