I'd say its definitely worth having serviced.
The Hamilton should be relatively straight forward as I'd expect a manual wind 17-21 jewel movement in it. I'm not sharp enough on Hamilton wristwatches to both ID it and say what what movement it is, but Hamilton used a few basic movements then. They are extremely high quality, easy to work on, and parts are still plentiful for them. Does it run at all if it's wound?
50 years ago, every watchmaker around could service Accutrons and other tuning fork watches(Accutrons are the best known of this type, but there were a few others). These days servicing them is a bit more specialized. They are high quality movements, but are different enough from both mechanical and quartz watches(and for the latter I'm referring to high end quartz that can be serviced) that you probably want to find someone who knows their way around them.
There are a few issues to be aware of. As mentioned, these were designed to use a 1.35V mercury battery, which is no longer made. Silver oxide batteries are available in the same size, but they are 1.55V volts. Some Accutrons run fine on the higher voltage, but some will occasionally or always "double index" which means that they will occasionally or or always run at double speed on the higher voltage. Some Accutron specialists(I use to talk regularly to a watchmaker who had worked for Bulova factory service during the heyday of the Accutron) would replace the coil and/or fork to one that would consistently run correctly at higher voltage, and although I was told this was a permanent solution, it was also not really sustainable one since you were relying not just on new old stock parts, but a small portion of NOS parts that basically that were shipped out of Bullova at the ends of QC tolereance to work properly at 1.35V.
I have been involved in old cameras far longer than I've been into watches(I've never been into wristwatches, but it's hard not to have some crossover on American wristwatches when you collect the sort of pocket watches I collect) and we run into similar issues with the mercury batteries. Many old camera meters are a relatively simple circuit of basically a photocell, galvanometer, and variable resistor that assume a 1.35V battery is present. We get further complicated by the fact that the most popular camera battery for this use was the PX625, which has a "flying saucer" shape and the only other chemistry I've ever seen it in is alkaline. That's a problem for any use that depends on consistent voltage, as alkaline voltage is very dependent on state of charge where mercury and silver both have flat discharge curves until near end of life. Some people offer a "spacer" to use the common LR44 sized battery in silver oxide and integrate a small zener diode(which will drop the voltage about .2V, or close enough that it can be calibrated out within the limits of what these meters can do) to bring the voltage down. Some put a zener diode inside the meter, then use a spacer ring(which you might be able to get away without in some cameras). Personally, I've always gone a different route-Zinc-air cells, which are super common as hearing aid batteries, also have a flat 1.35V discharge curve. A type 675 battery(blue tab) is about the same size as an LR44 cell. One company, Wein, a maker of various photo odds and ends sells these fitted to a spacer ring for about $10 each, a price which kills me considering that an 8 pack without the spacer isn't much more than that at Wal-Mart. Zinc-air has an issue compared to both mercury and silver, though. Camera meters are very low drain devices, and typically are used intermittently even on what might be considered a heavily used camera. Both mercury and silver batteries can last a decade or more in them-heck I've bought cameras(although it's increasingly less common now than in the mid-2000s when I got into this) that still had perfectly good mercury batteries in them. Basically, in most uses, the cell's self disharge rate is higher than the actual drain on the battery. Zinc air cells ship with a plastic tab covering air holes on the cell, and they are dead until that tab is removed. Once they start reacting with air, they have a finite life whether being used or not. Many state a month, although in a camera I've managed to get 6-8 months out of them before, especially if I covered some of the air holes back up. I'm not sure if this is viable for Accutrons, but it is a solution out there if you don't mind changing batteries.
Last thing, there are reports scattered around that mercury batteries are still made in Russia in both the PX625 size and the PX400 size, which is what Accutrons use. I ordered some of both about a decade ago, and they were indeed as advertised. I made a few bucks selling the Accutron batteries at watch shows, although even then factory in shipping and buying bulk I was paying $8-9 each(I sold them for $14 each and sold out in a few hours). I still have one or two of the camera batteries around-I think there's one in my Canon F-1n-although I ended up reselling most of them to a local camera store. Looking around, they seem to still be available, although not from the source I've bought from in the past(that is now just a reference page). I'm sitting here thinking now about ordering some more now...
