Hi BarbieChic,
The answer is no. Making an additional backup will not override your current backup.
As for passing down your old phone to your son, we recommend following the steps in the link below.
Sell, give away, or trade in your iPhone - Apple Support
To perform the recommended troubleshooting steps via the link above, complete the steps in the section below.
Before you sell, give away, or trade in your iPhone, see the Apple Support article What to do before you sell, give away, or trade in your iPhone, iPad, or iPod touch, and be sure to perform the following tasks:
* If you paired an Apple Watch with your iPhone, unpair your Apple Watch. (See the Apple Support article Unpair and erase your Apple Watch.)
* Back up iPhone. If you replace one iPhone with another, you can use the setup assistant to restore the backup to your new iPhone.
* If you’re replacing your iPhone with another one you have on hand, you can use extra free storage in iCloud to move your apps and data from one iPhone to another. Go to Settings
> General > Transfer or Reset iPhone > Get Started, then follow the onscreen instructions.
* Sign out of iCloud and the iTunes & App Store. Go to Settings > [your name], scroll down, tap Sign Out, enter your Apple ID password, then tap Turn Off.
* Erase all content and settings that contain personal information. If you previously turned on Find My for your iPhone, Activation Lock is removed when you erase iPhone, making it ready for a new owner.
We recommend the steps above because they will show you what to do before giving your iPhone away to your son.
Please do not hesitate to reach back out to us if you have questions or need any additional support.
Thanks for using the Apple Support Communities.
Cheers!