Sorry I didn't get to your earlier email about this.
We were in Big Bend National Park from October 13-18 last year.
I don't know where the National Park Service weather reporting station is, but if I'm guessing it's the Chisos Basin. The Basin is milder during the day than the surrounding desert, but I don't think it gets as cold at night there as it does on the desert floor -- at least, not in October.
During our trip, the weather ran the gamut, depending on time of day and our location in the park. We didn't hike the rim, as you are planning to, but can attest to the fact that the Chisos Basin is at a much higher elevation than Rio Grande Village and the surrounding desert floor. Daytime temperatures were milder up in the Basin than they were in Rio Grande Village where we camped. However, we still got plenty hot hiking The Window Trail from the overlook to the pour-off in the Chisos Basin -- the sun is unrelenting, and water's always a must.
I don't remember the exact temperature spread, but am fairly certain it was at least 80 degrees in the Basin and may have hit 90 degrees in Rio Grande Village at times during our stay. The temperature variation between the two locales isn't necessarily 10 degrees, I'm just illustrating you can expect a wide variation. Expect sudden weather changes, as well. We had a cold front blow in during our stay, and while daytime temperatures took a mild drop, the winds really whipped up at night and the temperature wasn't much more than 43 or 45 degrees. The winds were intense, and made us worry for the integrity of the pop-up trailer, at times (we were newbies to a pop-up, so some of the fear was probably unwarranted). We did not get any rain, and in fact only saw clouds on the very last day of our week-long visit. I'd almost forgotten what clouds looked like, and even those that did arrive were so small and high they looked like someone dropped cottonballs in a swimming pool.
We had a splendid trip, and managed the temperature and weather changes well I think. Just gauge what you may need to bring and be prepared for by making allowances +/- the averages reported by the Weather Service and NPS for the region you'll be staying in. Big Bend's a huge place, so the weather in one quadrant can be quite different from the weather in another on the opposite side of the park.