You can use \protect\phantom{$\cdots$} to be able to use a \phantom as desired:

As per egreg's comments you can also use \hspace*{2em} to specify much space you want to add. You can specify an absolute horizontal space as in \hspace*{1.0cm}, but it is better to use measurements that adjust based on the current font -- 1em is approximately the horizontal width on the letter m. Other useful horizontal spaces you can use include:

There is also a negative thin space, \! which is useful in certain situations.
Notes:
If you need more info on horizontal spacing refer to What commands are there for horizontal spacing?.
If you really need exactly the space used by $cdots$, you can use \settowidth to measure that size for you:
\newlength{\WidthOfCdots}\settowidth{\WidthOfCdots}{$\cdots$}
and then using
\hspace*{\WidthOfCdots}
will produce exactly the desired horizontal space.
Code:
\documentclass{article}
\usepackage{algorithm}
\usepackage{algorithmic}
\newlength{\WidthOfCdots}\settowidth{\WidthOfCdots}{$\cdots$}
\begin{document}
\section{What you have:}
\begin{algorithm}
\caption{ TestAlgorithm \newline
\textbf{SomeInputs} input1, \newline $\cdots$ input2, input3, input4, \newline $\cdots$ input5, input6, input7 \newline
\textbf{AnOutput} output1 }
\label{algo:TestAlgorithm A}
\begin{algorithmic}[1]
\STATE $oh yeah!$
\end{algorithmic}
\end{algorithm}
\section{What you want:}
\begin{algorithm}
\caption{ TestAlgorithm \newline
\textbf{SomeInputs} input1, \newline \protect\phantom{$\cdots$} input2, input3, input4, \newline \protect\phantom{$\cdots$} input5, input6, input7 \newline
\textbf{AnOutput} output1 }
\label{algo:TestAlgorithm}
\begin{algorithmic}[1]
\STATE $oh yeah!$
\end{algorithmic}
\end{algorithm}
\begin{algorithm}
\caption{ TestAlgorithm \newline
\textbf{SomeInputs} input1, \newline \protect\phantom{$\cdots$} input2, input3, input4, \newline \hspace*{\WidthOfCdots} input5, input6, input7 \newline
\textbf{AnOutput} output1 }
\label{algo:TestAlgorithm B}
\begin{algorithmic}[1]
\STATE $oh yeah!$
\end{algorithmic}
\end{algorithm}
Comparison of horizontal spaces:\bigskip
\begin{tabular}{ll}
\verb|,| & "," \
\verb|\thinspace| & "\thinspace" \
\verb|\enspace| & "\enspace" \
\verb|\quad| & "\quad" \
\verb|\qquad| & "\qquad" \
\end{tabular}
\end{document}
\documentclassand the appropriate packages so that those trying to help don't have to recreate it. – Peter Grill Oct 09 '12 at 21:00\protect\phantom{$\cdots$}. – Peter Grill Oct 09 '12 at 21:01\hspace*{2em}or similar for getting a non disappearing space at the beginning of a line. – egreg Oct 09 '12 at 21:04\:in LaTeX? – Werner Oct 09 '12 at 22:34