Q:

How to make a planet with differential gravity and constant daylight

I am hoping to create a planet with a greater rotational speed, making it oblong, thus causing differential gravitaion from the equator to the poles. I am hoping for the gravitarion to be lighter than the moon (possibly just heavy enough to prevent a person from jumping off the planet [the tangential speed would factor into this]) at the equator, and heavier than Earth's gravity (possibly by a lot) at the poles. I am hoping the difference can be relatively substantial, but I'm not sure how that would work, exactly. If the shape of a planet based on the various factors of rotational speed, mass/gravity, etc. can be obtained (a graph, like y = x^2), that would be most helpful.
I am also hoping to have the planet be "always day", such that there is always a relatively large amount of light impacting the entire surface. I considered having just the rotational speed account for this, but that wouldn't work with the necessary mass of the planet. The rotational speed could help factor into it, though. It was suggested to me that the atmosphere be denser, thus refracting the light from the star, which could possibly allow a person to see the sunrise and sunset at the same time (nifty). Plus, with the change in density, there would be a change in pressure, thus creating differences in planetary chemistry that would be interesting to explore. And as buoyancy is determined by density (I believe), it could result in certain elements floating a ways up in the atmosphere (a literal layer of water floating in the air sounds intriguing). As a last resort, multiple stars can be used. I'm hoping to simply "manipulate" the light, but a multi-star system could work if it would be the best possible option.
After the proposed rotational speed, mass, etc. has been presented, any additional effects they would cause on the planet would be very interesting, and greatly appreciated.

A:

It is certainly possible to have a planet with Moon-like gravity at the equator and heavier-than-Earth at the poles, if it rotates fast enough.
Unfortunately, figuring out exactly what shape a planet of a given mass would have at various rotational velocities, and what the gravitational force curve looks like across the surface, is rather complicated. If you're not too afraid of math, you can check out this lecture on calculating the equilibrium shape of the Earth for an example. It particular, it depends on the internal structure of the planet, and above certain critical values of angular momentum, there are actually multiple solutions, with rapidly-spinning bodies potentially taking on some pretty funky multi-lobed shapes. Fortunately, however, a simple oblate ellipsoid is always one of the potential solutions (as long as the rate of spin is low enough that the planet doesn't fly apart completely, anyway), and there are ways to simplify the problem. Mesklin, for example (already mentioned in the question comments) is assumed to have a degenerate-matter core, so the majority of its mass is concentrated near the center, and you can approximate the correct solution by ignoring the effects of deforming the distribution of the rest of the planet's mass away from spherical.
If you do something similar with your world, then we can figure out its shape given a certain mass and rotational period, and the gravitational force as a function of latitude, by finding the equipotential surface for the combination of centrifugal and gravitational potentials.
The effective potential is $U(r,\theta) = -\frac{1}{2}(\omega r \cos\theta)^2 - \frac{GM}{r}$, where $r$ is radius, $\theta$ is latitude, and $\omega$ is angular velocity. Setting $U$ to a constant value and solving for $r$ is kinda gross (it's a cubic equation), so you'll probably want to use a graphing program to figure out the exact shape numerically, but once you've got that, it's easy to calculate the gravity at the poles vs. the equator, and you can work backwards from there guessing-and-checking until you get a combination of polar gravity, equatorial gravity, and rotation rate that you like.
The "always day" bit is trickier, and hinges on just what counts as "always" and "day". Is it sufficient, for example, if "always" means "for a few hundred years at a time in one hemisphere" (plenty of time and space for a story to take place), and for "day" to mean "consistently illuminated at least to the level of a well-lit interior room, but with variations in brightness and heat above that point permitted"? In that case, I'd just put this planet in a system with a distant companion star on a long-period orbit highly inclined to the plane of the planet's orbit around its primary sun; that way, the companion star can provide constant illumination over one whole hemisphere, so there's never a real "night", on top of the seasonal and daily variations in illumination at any given point due to the primary sun.
Another option might be to make the world a rogue planet near an active galactic nucleus (quasar), which could provide habitable levels of insolation at several light years distance, with millenia-long orbits. It's the same general idea, just replacing the companion star with something bright enough to do the whole job of warming the planet all on its own, eliminating the need for another sun with its own superimposed illumination cycles, while still maintaining a sufficient distance to make the period of constant summer daylight seem like forever. Quasars, of course, have the downside of turning on and off rapidly and somewhat unpredictably, and they tend not to live in very nice neighborhoods, but the purposes of a good story I'd be willing to believe that, somewhere in the universe, there's a quasar that just happens to have had sufficiently steady output for sufficiently long to make this one unique world possible.

