Both are correct. To "weather with time" or "weather over time" suggest a gradual, continuous process. There's no exact moment at which the stone becomes "weathered." To "weather in time" almost suggests that someone is waiting for the stone to become weathered. There is a feeling of something happening at a definite time. For example, "names carved in granite last longer than names carved in slate, but even the hardest stone will weather in time."

“Weather over time” has become more common. Here is a Google Ngram. https://books.google.com/ngrams/graph?content=weather+in+time%2C+weather+over+time%2C+weather+with+time&year_start=1800&year_end=2000&corpus=15&smoothing=3&share=&direct_url=t1%3B%2Cweather%20in%20time%3B%2Cc0%3B.t1%3B%2Cweather%20over%20time%3B%2Cc0%3B.t1%3B%2Cweather%20with%20time%3B%2Cc0