Home Button Pressed when in an app:
Most apps will save their state (any bit of data that would be needed to keep itself consistent when being resumed), and the device will switch to the launcher app.
Home Button Pressed when on the Home screens:
This is configurable by many 3rd party launchers; however a lot of them default to bringing you back to your "main" home screen (the default launcher does this).
What your friend might be concerned about:
Switching apps can be memory intensive and cause several disk writes to save the data. This uses battery; however it is usually not the main cause of battery drain on devices (the operating systems are optimized to do these things).
Some applications will continue to run as a background application, so pressing the home button will not kill them (e.g., Google Play Music will still play after pressing the home button).
Your friend might be concerned that by pressing the home button you did not exit the program, and therefore it is still running. Android will unload those applications as necessary.
You and your friend should be more concerned with the display - which uses the most amount of power.