My daughter will be studying in the United States on a J1 visa. After her course finishes she has 30 days left on her J1. Can she then stay for a while longer to travel using ESTA?
Yes, your daughter can travel within the United States for 30 days after completion of her program. This is not "using ESTA", this is a grace period granted by USCIS for people who have completed their J1 program. From Adjustments and Extensions:
Travel Grace Period
Following the completion of their program, the period defined on the Form DS-2019, the United States Citizenship and Immigration Services (USCIS) allows participants a 30-day travel period commonly referred to as the "Grace Period." During this 30-day grace period, participants are no longer in J-visa status, and are under the jurisdiction of the USCIS. The USCIS grants this period to allow participants to settle their affairs and to prepare to return to their home countries. Program participants may no longer continue and/or complete exchange activities, nor may they work. Although participants may travel in the United States, it is recommended that they do not travel beyond the borders of the United States as they may not be permitted reentry.