If you have somebody that has a business asset that they have been depreciating and they want to take their name off the title and just let the other person that was on the title have it. Can the other person depreciate the asset?

Well, that wasn't all the answers. Let's see if we can restate this in perspective:

There is a Personal Vehicle used for business, and it is being transferred from Joint Ownership to Individual.

The same individual that uses the truck for one business will now use it for a different business? Or, the two of them had one business together, but now it is Sched C for only the son? Or???

So far, it's too vague.

What you need to consider is how the truck was owned legally and for tax purposes. It seems the Truck is being Gifted to someone, and removed from the original business? Or, being Sold to the other person, and being used in the original business? Or, a partnership is spinning off the Asset of value to one of the two partners?

You cannot just give away a truck without some consequence for taxes, personal finance, etc. And is there also a vehicle loan?

And if it is a different business, then you need to address that this is not a carry-on of the previous business. FMV at the time of retitle, for instance. Gift Tax consideration, or partner reconciliation against equity, perhaps.

You might want to get local mentoring on this. You see how many questions and variations will be involved. It's hard to have a discussion over the internet about these types of things you should be learning.