You can do a day hike up to Lone Pine Lake from the Whitney Portal. No permit needed for that.
If you're there for a few days do Lone Pine Lake in the afternoon and hang out at the wilderness center/permit office in the mornings trying to get same day or next day permits. You have a good chance of getting the permit depending on what day of the week it is.
Is a permit required to hike up to Meysan Lake? That would be another nice hike out of the Whitney Portal. I'd tend to say to keep the campsite. There's plenty to do in the area.
