There's not really any place super close, so you end up with four basic options.
1. If you arrive a few hours early but don't want to burn money on a hotel room that you'll only occupy for a short time, some people will just go ahead and park on the road near the museum, nap a couple of hours, then go ahead and start. I did that once and had no problems, but I was in an RV, so I didn't feel like people could come peek in the windows...
2. If you have a camper or a big enough car, you can park in the Walmart parking lot on Ramon overnight for free. I've done that when I've had our RV out there overnight while attending a swim meet my kids were in. Nothing there but a parking lot, but it is free.
3. There are lots of little, old but clean motels that start as low as $39.00 a night. I've done that as well, and they're within a couple miles of the museum.
4. If you want to camp somewhere nice, I'd recommend the Whitewater preserve.
http://www.wildlandsconservancy.org/pre ... water.html
It's about 15 miles away, but if you wanted to stay a couple of days, the camping is free and it's a very nice place. Tents only, and you have to arrive before 5:00 pm. The gates close from 5:00 p.m. to 8:00 a.m. but you can park your car outside the gate with no problem if you're leaving for an early start. BTW, I said free but they do accept donations, so that's up to you. Incidentally, the PCT is accessible from the preserve, and if you go south about 3 miles up into the hills there are usually beautiful fields of wildflowers in late March to April.
Hope one of those suggestions helps.
z