You get two chances at snagging a permit each day. When people call in and cancel their permits, they re-issue them in the morning. When people simply don't show up to claim their permits, they re-issue them in the afternoon. Show up early, not "near mid-day" so you will have a chance at both "lotteries."
There are literally hundreds of peaks worth climbing in the Sierra. Many have trailheads within 90 minutes of Lone Pine and have reasonable approaches. However, some of them are more technical than what I think your family would enjoy. Class three can be pretty scary when you're a day's hike from the car, you're above 13,000 feet, and one route-finding mistake brings the difficulty to class four or worse. (LeConte is a perfect example of this.) I would study R J Secor and Summitpost for ideas and compile a list of possible trips using several different trailheads. Print out a map and route descriptions for each trip.
I'll repeat my suggestion that you do Dana as a day trip while you're here. I think Agassiz would be perfect for your family. Nearby Mt. Goode is also worth considering. With the exception of LeConte, all of the suggestions in the above posts are good ones. Lone Pine Peak has a class two route from the Meysan Lake trailhead.