When several environments share the same counter -- as is the case for theorems, lemmas, corollaries, propositions, and definitions in the siamltex document class --- cleveref needs some help in order to figure out which one of the possible environments is actually being used for a given label.
The easiest solution to your problem, I believe, is to load the ntheorem package explicitly and then to re-run all of siamltex's definitions of its five theorem-like environments. (Naturally, if you don't need to use all five environments in your paper, you can skip redefining those environments you don't employ.)
The following is a modified form of your MWE that uses this approach.
\documentclass[final]{siamltex}
\usepackage{ntheorem,cleveref}
\makeatletter
% The next few lines are from 'siamltex.cls', but now with
% \renewtheorem{...} instead of \newtheorem{...}
\if@onethmnum
\renewtheorem{theorem}{Theorem}
\renewtheorem{lemma}[theorem]{Lemma}
\renewtheorem{corollary}[theorem]{Corollary}
\renewtheorem{proposition}[theorem]{Proposition}
\renewtheorem{definition}[theorem]{Definition}
\else
\renewtheorem{theorem}{Theorem}[section]
\renewtheorem{lemma}[theorem]{Lemma}
\renewtheorem{corollary}[theorem]{Corollary}
\renewtheorem{proposition}[theorem]{Proposition}
\renewtheorem{definition}[theorem]{Definition}
\fi
\makeatother
\begin{document}
\section{In the beginning}
Here's a definition.
\begin{definition}
\label{def:somedef} Some definition.
\end{definition}
If I use \textbf{cleveref} to try to refer to the above, I now get ``\cref{def:somedef}''.
\end{document}

Addendum As Ahmed Musa has pointed out in his answer, the code I copied from the siamltex.cls file (what's sandwiched between the \makeatletter and \makeatother instructions) seems unnecessarily complicated. For the sake getting cleveref to recognize the variants of the basic theorem environment, all that's actually needed are the following four instructions:
\renewtheorem{lemma}[theorem]{Lemma}
\renewtheorem{corollary}[theorem]{Corollary}
\renewtheorem{proposition}[theorem]{Proposition}
\renewtheorem{definition}[theorem]{Definition}