Different functions file for each site in multisite installation?

I’m trying to create two (for now) sites in multisite WP installation that use same theme, but require somewhat different functionality i.e homepage has a bit different layout and there will be some CSS changes.

Is there a way that I could use to override functions.php and other theme files for each site?

I would recommend a Parent/Child theme for this. Read the section linked below for a good example of making functions within your theme pluggable.

http://codex.wordpress.org/Child_Themes#Using_functions.php