Sorry no one replied sooner. <OBJECT>tag code inserted into the editor in a post or page is stripped out by Elgg code (so is any <SCRIPT>) as a security measure. So, try using one of the plug-ins designed for inserting video. I'm using Elgg 1.5 with the latest izap_videos plug in and finding that to be a user-friendly method -- the user only has to provide the URL to a YouTube video, and does not use the EMBED code given on YouTube. Using the URL alone this plugin will place a video player into a page or blog just like you want.
I regularly embed objects in a page I am using a version of the tinymce editor and it doesn't strip out the code I think its called yafftmce
Sorry forgot the link works great with tinypics etc http://community.elgg.org/pg/plugins/lockedoutuser/read/19693/yet-another-full-featured-tinymce