Overview:
A nice campsite called Eagle Spring lies on the east side of Mission Peak, which is far better known for its day-hiking.
Why bother camping near the peak? Because the best times to be at the summit are sunrise and sunset, and it's much easier to get up there after you've set up camp and most of the climbing is out of the way.
This bundle of Mission Peak day-hikes shows the best starting points for an Eagle Spring campout. The campsite is very easy to find: just look for the privy, spigot and picnic table near the perimeter trail below the east face of the peak.
The campsite must be reserved. Call (888) 327-2757, press option 2, then 1. Fee is $5 per night.
Best times to go: Spingtime is ideal, though you may have to reserve your spot several weeks in advance (or go during the week). Summer and fall are best avoided because of wildfire risk. Winter could be quite an adventure, especially if there's snow in the forecast. Just make sure you're equipped for severe weather.
One interesting option: have a friend drop you off at Ohlone College and pick you up at Ed Levin County Park. Another possibility: car shuttling, leaving cars at Sunol Wilderness and Stanford Avenue.
Tips:
Backpacking requires proper gear and the physical stamina to carry it - this is especially true while hiking up to Mission Peak. While Eagle Spring is a great place for first-time camp-outs, make sure you're up for it before launching an outing.
Parking options:
The best option is Sunol Wilderness, where it's safe to park your car overnight - though you will have to pick up an Ohlone Wilderness Trail permit ($2 at the Sunol HQ).
If you go up via Ed Levin County Park or Ohlone College, you'll need to be dropped off or picked up; overnight parking isn't allowed.
Ohlone Wilderness Trail starts at the Stanford Avenue entrance to Mission Peak Regional Preserve. Be aware that the area is prone to car break-ins, so don't leave anything of value in your car.