Interesting that you should mention that. Alan Scott’s daughter, Jade, inherited his Green Lantern energy powers. She briefly lost them and gained the power to manipulate plants, but then her boyfriend at the time, Kyle, restored her powers with his ring. Whether her current powers are powered by Alan’s or Kyle’s energy is unclear, though I’m of the opinion that Kyle merely jumpstarted her latent ability rather than supply her with a new one.
Her powers are innate, so she never has to recharge them, and they don’t have any weaknesses either.
Alan’s weakness to wood always seemed a bit goofy to me, but DC recently introduced a similar weakness to Powergirl, who is now vulnerable to raw or unprocessed materials. Shoot her with a particle gun gun and you’re out of luck, but poke her with a sharp stick and you can do some serious damage.
GL’s “weakness to yellow” was all a part of the silver age approach to comics at DC, and to a lesser extent, at Marvel*. If a character were powerful enough, he or she had to have a weakness that the bad guys could exploit. Superman had kryptonite, and it was apparently available in corner drugstores and by mail order through Sears, it showed up so often.
GL had the color yellow, which makes sense in silver-age comics logic. Weaknesses were frequently related to powers; Superman’s powers came in part from yellow solar radiation and the lesser gravity of earth, so his weakness was radiation from parts of his home planet. GL’s power was a specific color, so another specific color was its weakness, in addition to needing to be recharged every 24 hours.
*At Marvel, it was usually the villains with the cosmic level powers and an exploitable weakness.