Part One of Two: Ring Produced Equatorial Ridges.
Possibly the innermost rings of the planet suffered orbital decay and fell down on the equatorial regions of the planet.
Iapetus, a moon of Saturn, has a high equatorial ridge part way around its equator. There are 4 main theories for the formation of the ridge, but none explains why it goes only part way around Iapetus. One theory is:
Iapetus could have had a ring system during its formation due to its large Hill sphere of ~49 Iapetian radii, and that the equatorial ridge was then produced by collisional accretion of this ring.
Whatever the origin of the equatorial ridgge of Iapetus may be, it might be possible for the inner rings of a habitable world to gradually fall on the equatorial regions of that world and build up a tall equatorial ridge.
And of course highlands running around the equator of a planet would have some effect on winds and air circulation patterns. Thus they might make the equatorial regions less stormy or more stormy, depending on meteorological factors.
Part Two: Ring Shadows.
Planetary rings are composed of many small solid particles. The ring particles are opaque. All light which strikes a ring particle is reflected or absorbed. None of the light which strikes a ring particle passes through that ring particle to the other side.
So light passes though rings through the spaces between ring particles.
If the distance between the inner and outer edges of your ring system is thousands of kilometers, The ring should block almost all the light which strikes the outer edge aimed at the center of the planet.
So when ever the equator of the planet and the ring system in the plane of the equator is aimed at the star, the equatorial region of hte planet will be in complete and total shadow. Light and heat can only reach the shadowed region sideways from the unshadowed region of the planet. Thus the region directly below the rings should be much darker and colder than regions outside the shadow, appecting wind patterns, air flow, and weather.
The more the ring system over the equator is tilted relative to the star, the more transparent it will be to the star's light, but it will also cast a wider shadow over the planet's surface. The ring system would probably be only tens or hundreds of meters - meters and not kilometers - thick, so if it was directly in line with the star it would have a shadow merely tens or hundreds of meters wide, which might not be enough for creating a major weather pattern.
So by deciding how large the spaces between particles are compared to the sizes of the ring particles, you should be able to calculate the angle between the rings and the star which will enable the rings to cast the widest and darkest shadow over the equatorial regions and make the largest effect on the planet's weather patterns, while still leaving the polar regions out of the shadow and in full star light.
Part Three: Origin and Age of the Rings.
I note that there is some uncertainty about the ages of hte various planetary rings in our solar system. Are they left over from the formation of the planets billions of years ago, or are they much younger and temporary features?
Estimates of the age of Saturn's rings vary widely, depending on the approach used. They have been considered to possibly be very old, dating to the formation of Saturn itself. However, data from Cassini suggest they are much younger, having most likely formed within the last 100 million years, and may thus be between 10 million and 100 million years old.
Based on current depletion rates, they may disappear in 300 million years.
But on Earth it took billions of years for life to arise and eventually produce an oxygen rich atmosphere breathable for humans and for large multicelled land plants and land animals to evolve. So if you want your planet to be interesting for most types of science fiction stories it should be billions of years old, and yet have planetary rings which might be only tens or hundreds of millions of years old and so be billions of years younger than the planet.
One way for planetary rings to form is a large object colliding with the planet and putting a lot of debris into orbit around the planet, which will form rings which might eventually form a moon of the planet or might stay as rings of gazillions of particles.
So you could have your billions of years old habitable planet have a ring system formed by a collision only tens or hundreds of millions of years aog, billions of years after the planet formed.
Except that if a large enough object to form planetary rings collided with the planet that should have destroyed the planet's atmosphere which took billions of years to form and also turn the whole planet into red hot lava. That should kill all life on the planet and set the clock back to zero, so it would take the planet billions of years more to become interesting for most stypes of science fiction stories, and by then the planetary rings should have been destroyed.
There are three possible ways to get around that problem.
One) After a collision formed the planet's rings, and the red hot planet gradually solidified and cooled, an advanced civilization found the planet and terraformed it to make it habibable. Thus the planet could be interesting for most types of science fiction stories while still having impressive planetary rings.
Two) The collision which formed the planet's rings was not a collision with the planet, but a collision between other objects. Possibly the planet had one or more natural moons orbiting in stable orbits. Then the planet accidentially captured another object which became another moon, and this captured moon had an orbit dangerously similar to that of one of the planet's preexisting moons. And after maybe many millions of years the captured moon and the preexisting eventually collided and shattered, forming the ring system.
Three) The rings didn't form from collisions, but from small objects left over from the formation of the planet. Presumably the rings would have been getting thinner and thinner over billions of years, so if they are still significant they would have been very impressive when the planet was first formed.