For the ultimate purpose of saving money, is it better to fill up a car’s fuel tank, or only buy enough to bring up to 3/4 or 1/2, etc?
The reasoning presented to me was that the additional fuel was extra weight that reduced the gas mileage. That’s true, but I’m not so sure it’s significant. Any ideas on what the answer is?
One also needs to account for the costs of the extra stops to buy fuel, but it’s not obvious to me how to quantify those.
That’s one of the more ridiculous things I’ve heard. The idea that carrying a few less gallons of gas will lessen your vehicle’s weight to the point of it being noticeably more fuel efficient is silly. Especially since, as you mentioned, you’ll do a lot more stopping and going when you’re having to fill up more frequently.
If saving a few pounds in the vehicle’s weight meant better fuel mileage, I assume people would fast before taking trips. 
We have done this one a few times and it doesn’t matter. Extra weight from the gas doesn’t really add much to the energy requirements. It will only add only a couple of 2% to the vehicles weight at most and that still does not translate directly into a couple of percent savings in fuel. Added weight is mainly important when accelerating and inertia takes care of it during the rest of the time. That is why tractor trailers get decent gas mileage all things considered especially because they weigh many times what a passenger car does. The extra gas stops would likely wipe out any miniscule savings it causes.
I think you are better off worrying about the price you pay for gas, rather than the amount you buy. Find the best price you can at a convenient location, and fill up there. Any savings from driving a lighter car will be lost if you have to buy gas at a more expensive gas station because you are low on gas.
When the price is high, but as little as you can without being silly about it. When the price is low, fill 'er up.
less gas in the tank increases the chance you will have to fill up at a high priced station, go out of your way for gas, or running out of gas, all of which cost money. Filling up more often even at gas stations along your route still costs a bit more for the slight diversion of route stop, start, accelerating that you don’t have to do if you have enough fuel to fill up another time.
Now for a temporary price spike you may save by not filling all the way up.
This is an interesting topic considering that an earlier thread on the lack of spare tires in cars suggested that a reason for this was to save weight and therefore increase gas mileage.
Here’s how to quantify it: Write down your odometer reading. For the next hundred gallons, buy only less than a whole tank. Write down your odometer again. Now, for this hundred gallons, be sure to fill up your tank every time. Look at your odometer again. Now, compare the readings: Did you go farther on the first hundred gallons, or the second hundred gallons?
(In both procedures, be sure to write down exactly how much gas you bought each and every time. To the thousandth of a gallon, if possible. Otherwise, you could end up comparing 101.345 gallons to 102.256 gallons, or whatever.)
The best way to save on gas is to buy it early in the morning. Cold gas is denser. A gallon bought in the morning will expand during the day. In Canada the pumps have to account for this variance, but in the US the feeling is that it’ll just average out and those in the know can benefit if they want.
Underground tanks don’t vary in temperature by more than a couple of degrees, and gas is fairly incompressible.