Think of the tidal effect that the sun and moon exercise on the sea and you'll be part way to an answer.

As far as i know gravity doesnt fluctuate it is a constant force from every object, what can change is the gravity at a point which is a sum of all the attractive forces from all other objects. And to your main question i think that there would be an increadiably small change because of difference in distance from the attracting body, though i dont think this would be noticable.

Assuming the Earth's radius is 6370 Km, the distance to the sun is 150 million Km, the Earth's mass is 6x10^24 Kg and the Sun's mass is 2x10^30 Kg, it's easy to work out that, at the Earth's surface, the gravitational pull of the Sun is about 0.0006 g. At the equator around the equinoxes this precisely adds to the Earth's gravity at midnight and subtracts from it at noon, so the difference that the Sun makes to midnight and noon gravity is about 0.0012 g (i.e. 0.12%).

Some additional complications arise from the elliptical orbit of the Earth around the Sun, the tilt of the Earth's axis and the fact that the Earth is a sphere.

In modern times the perihelion (147 million Km) occurs around 3rd January and the aphelion (152 million Km) occurs around 4th July, so the Sun's gravitational pull is about 6.9% more in January than in July and the daily variation in gravity will itself vary by that amount.

The tilt of the Earth's rotational axis combined with the lattitude dependent distance to the axis will also produce variations in the daily change of gravity.

Similar effects also apply to the Moon except that its gravitational influence is somewhat greater (on average about 0.022 g or 2.2%) and its orbital eccentricity is greater giving rise to a variation of some +/-12% in this figure.

To summarise, a maximum day/night variation in gravity of about 2.5% occurs at the equator and almost all of this is caused by the Moon's gravitational influence.

2.5% seems an incredible variation, but I can't fault your calculation, Chris.

If the variation in gravitational pull affects the weight of the atmosphere too, then presumably there is a corresponding 2.5% variation in barometric pressure over a lunar month, as well. That's 25 millibar, or a couple of centimeters of mercury. That should be easily observable statistically ?

In fact, a variation on that scale should have a material and detectable effect on a lot of processes, including ballistics, the speed of elevators, the friction of my tyres on the road, and the blood pressure in my brain. Maybe lunacy is not a myth after all.