Best How To :

The escape_javascript function is on the rails side. You don't want it in the output so it'd be within the #{}.

And you threw out most the quotes. Still need those.

Hopefully there's no typos but a direct translation should look more like this:

-if @discussion_category
  javascript:
    $("#{escape_javascript(dom_id(@discussion_category))}_notice").html('');
    $("#{escape_javascript(dom_id(@discussion_category))}")
      .html("#{escape_javascript(render(partial: 'display_change', locals: {discussion_category: @discussion_category}))}");
    $('#discussion_category_links').show();

When building messy RJS stuff like this it can be much cleaner looking to use the j alias for escape_javascript, which would help in narrowing down syntax issues as well:

javascript:
  $("#{j dom_id(@discussion_category)}_notice").html('');
  $("#{j dom_id(@discussion_category)}")
    .html("#{j render(partial: 'display_change', locals: {discussion_category: @discussion_category})}");

Finally, there's probably no reason to escape the output of dom_id, meaning you could just:

javascript:
  $("#{dom_id @discussion_category}_notice").html('');
  $("#{dom_id @discussion_category}")
    .html("#{j render(partial: 'display_change', locals: {discussion_category: @discussion_category})}");

... which is beginning to look a lot more readable.