I did this to my website and it works a treat. This applies if you are using the default BP theme, or a child theme. You will need to go to wp-content/plugins/buddypress/bp-themes/bp-default/header.php
At approximately line 47, you will see the opening tags <div id="header"> Look to about line 52, and you will see the code for the home tab. Replace the code with this
<?php if ( 'home' != bp_dtheme_page_on_front() && bp_is_active( 'home' ) ) : ?>
<li<?php if ( bp_is_page( BP_ACTIVITY_SLUG ) ) : ?> class="selected"<?php endif; ?>>
<a href="yoursite.com/activity" title="<?php _e( 'Home', 'buddypress' ) ?>"><?php _e( 'Home', 'buddypress' ) ?></a>
</li>
<?php endif; ?>
Remember to add http:// before the yoursite.com/activity, for some reason it doesn’t show when I type the full URL in here.
If you look below that, you should see the code for the activity tab, delete this code. You should now have removed the activity tab from the menu, and the Home tab has now been changed so it directs to the activity stream. Any problems message me.