Here is the new monster:

package IO::patch;

# The following code:
#    use CGI;  my $q= CGI->new();  open CGI, "< $0";
# gives:
#    Can't locate object method "new" via package "IO::Handle"
#
# But download this code and put it into lib/IO/patch.pm and
# either add "use IO::patch;" to your script or add
# PERL5OPT=-MIO::patch to your environment, and this problem
# goes away.
#
# Only class methods are subject to this problem.  This version
# of IO::patch even handles class methods that are AUTOLOADed,
# but doesn't trap class methods that have the same name as
# methods (or routines) in IO::Handle except for new().

require IO::Handle;

use vars qw( $AUTOLOAD );

my $oldNew= \&IO::Handle::new;

*IO::Handle::new= \&newNew;

*IO::Handle::AUTOLOAD= \&AUTOLOAD;

sub newNew {
    my $pkg= $_[0];
    if(  UNIVERSAL::isa( $pkg, "GLOB" )  ) {
        my $name= *{$pkg}{NAME};
        my $can= UNIVERSAL::can( $name, "new" );
        require Carp   if  $^W;
        if(  $can  ) {
            Carp::carp(
                "File handle vs. package name conflict ($name)"
            )   if  $^W;
            $_[0]= $name;
            goto &$can;
        }
        Carp::carp(
            qq(Called new() on an open IO::Handle or can't locate)
            . qq( object method "new" via package "$name")
        )   if  $^W;
    }
    goto &$oldNew;
}

sub AUTOLOAD {
    my $pkg= $_[0];
    my $meth= $AUTOLOAD;
    require Carp;
    if(  UNIVERSAL::isa( $pkg, "GLOB" )  ) {
        my $name= *{$pkg}{NAME};
        $meth =~ s/^IO::Handle:://;
        my $can= UNIVERSAL::can( $name, $meth );
        if(  $can  ) {
            Carp::carp(
                "File handle vs. package name conflict ($name)"
            )   if  $^W;
            $_[0]= $name;
            goto &$can;
        }
        $can= UNIVERSAL::can( $name, "AUTOLOAD" );
        if(  $can  ) {
            eval { require Devel::Peek }
                or  Carp::croak( 
                    qq(Can't AUTOLOAD method "$meth" via package "$nam
+e")
                    . qq( unless you install Devel::Peek\n)
                    . $@
                );
            my $gv= Devel::Peek::CvGV( $can );
            {
                my $alPkg= *{$gv}{PACKAGE};
                no strict 'refs';
                ${$alPkg."::AUTOLOAD"}= $name . "::" . $meth;
            }
            Carp::carp(
                "File handle vs. package name conflict ($name)"
            )   if  $^W;
            $_[0]= $name;
            goto &$can;
        }
        Carp::croak(
            qq(Can't locate object method "$meth" via package "$name")
        );
    }
    Carp::croak(
        qq(Can't locate object method "$meth" via package "IO::Handle"
+)
    );
}
[download]

I've tested it and it works well in lots of situations. The code could probably be drastically cleaned up, but that is less important than getting this out there.

Update: Thanks, bbfu, fixed.

        - tye (but my friends call me "Tye")