--stringparam should allow string parameters that contain both single quotes and
double quotes.  XPath strings can contain both; it's just that the current
implementation of --stringparam is stupid and complains if the parameter
contains both kinds of quotes instead of correctly wrapping the parameter in an
XPath string expression.

Other information:
I suggest the following way of wrapping string parameters; it handles all
combinations of quotes correctly at a minor penalty in performance.  Replace
each single quote in the parameter with the seven-character string ',"'",' . 
Then add concat('',' at the beginning and ') at the end.  (The extra '' at the
beginning is needed so that concat gets the required two arguments when the
input has no single quotes.)

I was using a bash script along the lines of
    xsltproc --stringparam foo "$1" grab-parameters.xsl dummy.xml
as an easy way to make a properly escaped XML file with elements containing
given text strings.  However, this didn't work if $1 contained both kinds of
quotes.  My workaround was to implement the technique described above in bash to
wrap $1 in an XPath expression and pass the expression using --param.  My script
now looks like this:
    apos="'"
    function pq {
        echo -n "concat($apos$apos,$apos${1//$apos/$apos,"$apos",$apos}$apos)";
    }
    xsltproc --param foo "$(pq "$1")" grab-parameters.xsl dummy.xml
Enjoy the workaround until/unless this bug is fixed.