Everything else being equal, lighter strings will produce more overtones in relation to the fundamental. Again, if everything else is equal, lighter strings have less tension. Less tension means more excursion. More excursion means more overtones.
How does Tension, or length, on a string effect overtones?
Many guitarists over the years have commented on how fat Tony Iommi's guitar tone is. Because of his prosthetic finger tips, he uses very light gauge strings tuned down a half step.
Please keep in mind that higher frequencies are easier to hear than lower frequencies are. So if a low B string didn't produce many overtones, it would be much more difficult to hear because the fundamental is low. The human ear "hears" the overtones and to an extent, it "fills in" the fundamental. Some bass builders will tell you this is why bolt on necks sound fatter.
You can test this yourself with a tuner by the way. Take a heavy string, tune it up to pitch and play your note. Now, take a lighter string and stretch it across the same scale length as the heavier string and tune IT up to the same pitch.
Watch how more quickly your tuner will "settle" with the heavier string. That is because it is "finding" the fundamental more quickly.