first off, what is the condition of the wiring harness in the car?
If it is the original harness, replace it with a new one.
Go to Ken Cates Hudson webpage and he has a nice write up on doing a 12v conversion.
https://hudsonrestoration1948-54.com/12v%20conv%20data%20ParkWaldrop.pdf

    First off, is the wiring on your car still original? If so, that is a major part of the reason for dim lights. Next is the terminals, they are probably corroded, or loose. Third, is your generator putting out the volts and amps is supposed to? Should be 7.5 volts and up to 30-35 amps. If not, then generator may need a rebuild, and then the voltage regulator needs to be checked. Even with excellent wiring, the best thing you can do to get bright headlights is to use a headlight relay under the hood. There is over 10 feet of wiring going from the dash headlight switch, down to the dimmer switch, and from dimmer switch to headlights, which means enough resistance to cause a voltage drop at the headlights. The headlight relay takes the power directly from the battery and shuttles it to the headlights in about 2-3 feet of wiring, as well as bypassing the contacts of both the headlight switch and dimmer switch. So, when set up well, the 6 volt lights can be bright as 12V.  As for your windshield wipers, they are vacuum operated, not electric, unless someone converted them to an electric wiper.
     With the above said, if you are committed to going to 12V, it is a relatively straight forward conversion. All lights must be changed out to 12 V. The instruments will work just fine as your 1951 does already have a constant 5.5V instrument voltage regulator on the back of the fuel &  temp gauge cluster. I had converted my 53 Hornet to 12V 36 years ago, and the same 5.5 volt instrument regulator is still doing it's job.  The 6V system is positive ground, and for 12V you will want to go to negative ground as the 12V alternators are negative ground due to a diode keeping the output at the BAT terminal + positive. This will present some more work, if you want to use the radio.......To hook up the alternator, the ARMature wire of the generator now connects to the BAT terminal on the back of the alternator. To make it simple, I suggest you use a 63 amp GM Delco Remy 10SI alternator, with three wires. The BAT terminal as mentioned before corresponds to the ARM terminal of the old generator.  You need to remove the old regulator, or at least the wires from it. The wire from BAT terminal of the Alternator now needs to be run to the battery side of the starter relay (solenoid) on the inner left fender. The former ARM wire is also connected to the battery side of starter solenoid, and runs back to the main circuit breaker behind the dash. The former field wire of the generator  is connected to the white field wire of the DA plug of the alternator and this runs back to be connected to the wire coming from the dash AMP light. The red "sensor " wire from the DA plug can be connected the BAT terminal on back of the alternator, but some people prefer to run it back to either the battery side of the starter solenoid, or to the main circuit breaker in the dash, for quicker charging response to a voltage drop from accessory draw. I do not care for the single wire alternator as you have to add a relay in order for the dash AMP warning light to work.
'

A lot to think about...appreciate your help!

Another advantage of converting to 12v is that you can use 12v LEDs in most of the sockets. 

I’ll give Ken respect for many more years of experience in Hudsons than I’ll ever have, but in working with other 6V systems, I’ve never been able to get 6V headlights as bright as12V halogen sealed beams. In theory it’s possible (Ohms Law and all that), but in practice, it doesn’t work as well.  In search of better light, I eventually bought Trucklite LED headlights to replace my halogen sealed beams.  That made a real difference in brightness and pattern.  They look different than sealed beams, but I’d rather have the better light at night. 

You can always install  a 6 volt headlight relay and that will brighten up the headlights quite a bit, and 6 volt Halogen headlights are available.

I will assume you have given this conversion some thought.  However, you should be aware that the results you seek may be available with six volts, and then you would not have to embark on a time-consuming conversion.

I think the main issue you are concerned with, is lighting.  (You mentioned "rain", which says "windshield wipers" to me.  But the Hudson's wipers are vacuum, so neither six nor twelve volts will have any bearing on engine vacuum!)

I currently run LED lights and headlights on my 6-volt 1937 (Terraplane), and I have also used halogen bulbs in the past.  Both are available in 6-volt (and for positive ground).  I did take the precaution of using headlight relays (to circumvent the problems of miles of ancient wiring and corroded switches throughout the system. I also grounded the headlight sockets to the car's frame.  Neither of these fixes is expensive nor particularly time-consuming.  Of course you need to make sure you have the adequate, larger-diameter 6-volt battery cables in your car, and they the system is properly grounded.  (Cable size is especially important for good starting.

Naturally, replacing the entire wiring harness would be ideal, but it's awfully expensive.  Instead, you could gradually work through the system, replacing bad wiring with new wire you buy at your local parts house.

If you'd still prefer to switch over to 12 volt, go right ahead.  I'm just saying that there are work-arounds by which you can still get satisfactory results with a six volt system.

The taillights and headlights are quite bright. 

You don't mention whether or not your Hudson has overdrive, which is also affected by a 12V conversion. I'm from the school of thought that 12V conversions create as many problems as they solve. Would follow Ken's advice re going through the existing likely tired 6V system first.

Tony, you need to figure out all the things you want to use the car for, then do a net plus and minus comparison.

If you want modern electronics in your car, then you almost have to go 12V. These would include, a better radio or audio system, USB outlets or adapters to charge your phone, or a 12V socket to plug in an adaptor, or to power your GPS.  If you are more interested in keeping it close to original, and you don’t drive it much or need lots of light at night, a 6V system will work fine. But the more you want to add electrical and electronic upgrades, the better 12V becomes.  

As for wipers, if you really need to see in the rain, switch to electrical wipers. The vacuum things were crappy even when new. And they’ve only gotten worse with age.  And it seems to me that the times I needed the wipers to work the most, is when I was accelerating to get into traffic or going up a hill, or passing an 18-wheeler in his spray wake; and that’s when the vacuum wipers would slow or stall, as engine vacuum dropped.  The operating principle is basically wrong.  

There are good engineering reasons all the manufacturers abandoned 6V electrical systems 60 years ago and vacuum wipers by 1970 or so.  

I think it's reasonable to say that the 12V electrical conversion [i]does not [/i]create as many problems as it solves. Frankly it solves a great deal. It's not a quick and easy swap though. It requires engineered solutions to a multitude of systems (charging, starting, gauges and electronics, electric overdrive, etc.). I know I, Walt, and a multitude of other Hudson drivers don't regret for a moment doing the conversion.

As Peter said, it's worth figuring out what you want to do with the car. Mine is a driver. I pride myself with how many miles I put on the car each year. Many would claim it's "less original" in its current state, but it's original enough to run across the East coast on the same engine and trans that came from the factory. I'm just able to enjoy it that much more easily. If I go on a road trip, I have air conditioning for my wife and dog. I can charge up my phone if I get stranded. If my wife takes the car out and the battery dies, I know she can get a jump from a stranger without damaging anything. I'm also able to go into pretty much any auto parts store and buy a battery or alternator or coil or bulbs when I'm on the road. My car starts in less than a second in the middle of winter and the lights are super bright at night.

If you're building a museum piece, maybe there are too many cons. But for me, it's all about driving my flathead Hudson as far as I can 🙂 Ken's site documentation is the place to start.

Oh, and PS: go electric wipers or don't drive in the rain. Like Peter said, the principle is backwards. But the vacuum motor is more original too.

What should we look for in a shop that has done this before?  There is a 1952 Hudson that I’m looking at and the owner was unable to complete the 12v conversion.  Are there any shops in Northern California folks have used to convert their Hudson yo 12 volts?

Jaime:  I doubt you can find a shop that will agree to take on an incomplete 12v conversion project.  Too much risk if the job doesn’t go as planned.  

If you can twiddle wrenches and use a volt-ohm meter, do it yourself with the help you can get here.  If you can’t do that, then don’t buy this project!