Can I give an Apple Watch to my son with cellular connection and it acts completely independent of my iPhone?

What series watch is it? If you simply give the watch to your son, you both will share the same phone number and many other features (messages, calanders, contacts, etc.). Does you son have an iPhone? If so, unpair the watch from your iPhone and set it up as a new watch on his iPhone:

Set up your Apple Watch - Apple Support

If not, unpair the Watch from your iPhone and use watch Family Setup (if it is an Apple Watch Series 4 or later with cellular, or Apple Watch SE with cellular, with watchOS 7 or later):

Set up Apple Watch for a family member - Apple Support

In either case see the 'Before you sell, give away, or trade in your Apple Watch' section in:

What to do before you sell, give away, or trade in your Apple Watch, or buy one from someone else - Apple Support

and you should contact your carrier to make sure they can support watch Family Setup (usually an additional $10 monthly charge):

Apple Watch - Carriers - Apple

Yes, but it has to be the Apple Watch SE + cellular, 2nd generation. I don’t know about previous gens. And it has to be on a service provider that provides the service. Spectrum does not provide this service for the Apple Watch SE + cellular.

The Watch needs to pair to an iPhone but once it is set up, and provider set it up with a phone number, it can have its own stand alone phone number for calls and text messages. It will also needs its own iCloud account.