Why only one day? There is so much to see.
I did a rim to rim hike a couple years ago that went like this:
Day 1, S. Kaibab Trail to Bright Angel Campground (Phantom Ranch).
Day 2, Still in Bright Angel Campground, did a nice day hike.
Day 3, Hike to Cottonwood Campground in early A.M. Drop gear at Cottonwood Campground, hike to Ribbon Falls, have lunch and hike back to Cottonwood Campground to set up for the night.
Day 4, Hike out the North Kaibab Trail.
On day 1, a buddy from work drove my car to the North rim and met us at Bright Angel Campground for the night. I had driven his car to the South rim.
We did this in October and there were no services open North of Bright Angel Campground. There was a water line break below Roaring Springs, so we had to carry a lot of extra water to Cottonwood Campground. We also purified some from Ribbon Falls to take back to camp with us and stay topped off.
We had heavy, cold rain on day 1, (Thank God for Gore-Tex) that cleared off by the time we got to camp, so we got to set up with no foul weather.
Day 2 was sunny and hot, shorts weather. We did a couple nice short day hikes not getting too far from Bright Angel Campground. Day 3 started out very brisk as it was ~4 in the morning when we packed up to depart and the weather was very pleasant throughout the day. Shorts in the afternoon. It got windy and cooler in the night and on day 4 we had rain until about Roaring Springs, then heavy snow the rest of the way to the North rim. The
North Rim was closed inbound at the time and we were beat, so we drove to the forest just outside the GCNP boundary and camped for the night instead of driving in the snow. We drove back to Flagstaff the next morning.
When I do it again, I will go North to South. The two main reasons are that the view will be different heading the opposite way, and the North Rim is ~1,000 feet higher than the South Rim and that makes a big difference on that last day.
There is a huge difference in hiking rim to rim in a day and really hiking the Grand Canyon. I would recommend the latter.
Dave