Question:

I drive the car registered in my wife's name.

She drives the car registered in my name.

It just happened out that way.

So, anyway, suppose I get a new car... the dealer takes the tags off my car (in her name) and does the registering for me.  Is there any way at this point to change the name without any hassle?  Or, because it is in her name will it be required to stay in her name on the new car?

Eventually, of course, when we replace hers, I would do the same.

I would say go to DMV and have both names on both registrations. I wouldn't worry about the plates; I'd worry about who is on the title. Worrying about who is named on a registration is worrying about who is entitled to the plates and nothing more.

A simialr situation to this just happened to me last week: Bottom line, you cannot change registration at the time of the transaction. You either come with the paperwork done with both names prior to the transaction, or get it done after, but it cannot be changed at the time of purchasing.
My wife ran a red light on the 26th, and t-boned a minivan with only the driver. The van flipped 1.5 times, and everyone walked away with nothing more than a scratch. My 9 month old daughter didn't even wake up from her nap in her car seat in the backseat. We purchased a used car 3 days after the accident and tried to get my name on the new car as well as her name: not gonna happen. It can be fixed later, but not during the transaction.
BTW: I do not work for the DMV nor was I compensated in any way for this commerical ad.

Why don't you just get new issued plates?

Unless you really, really want to keep the old tags, that is the easiest way to get your name on the new registration.

Quoted: Why don't you just get new issued plates? Unless you really, really want to keep the old tags, that is the easiest way to get your name on the new registration.

It depends on when I get the car.  I still have more than a year on my current reg.  I think they credit that time when you transfer the plates.

On a two year registration the DMV will issue a refund if you cancel the registration during the first year. No refund is given for any unused period in the first year.