How can one create an element with a name that is not determined until
runtime? I tried assigning to a variable and putting the variable in the
name attribute of xsl:element, but that doesn't work, and I tried using
the name function within the name attribute, since the name of the element
should be the name of another element, but that doesn't work either.

It would be easier if you've showd us a code snippet.
I guess you didn't take into account that the name attribute
of xsl:element doesn't take an XPath expression. You have to
use an AVT:
 <xsl:variable name="foo" select="'bar'"/>
 <xsl:element name="{$foo}"/>