If that family holiday is in the near future, the person won't be able to legally enter the USA.
Anyone with a criminal conviction (or, indeed, anyone who's ever been arrested - even if they were completely innocent) is not eligible for the Visa Waiver Program and must apply for a visa.
Part of the visa application process involves attending an interview in London. The US Embassy website states:
"The average Appointment Wait Time for all nonimmigrant visa categories is generally 30 to 40 days. "
http://www.usembassy.org.uk/cons_new/visa/niv/ appointment.html
Additionally, anyone with a criminal conviction has to answer 'Yes' to Question 38 on the visa application form. The US Embassy website states:
"Applicants who tick yes to box 38 should be aware that it is difficult to predict how long the application will take to process, but can expect to wait a minimum of 14 to 16 weeks before being advised of the outcome of their application. Applicants who are advised at the time of the visa interview that their application will require additional administrative processing should allow at least 60 days for this stage of the application to be completed. Please note that the processing times quoted are approximate and cannot be guaranteed."
Full visa details are here:
http://www.usembassy.org.uk/cons_new/visa/niv/ apply.html
The application fee is $100 if you're invited to take part in the trial scheme that let's you pay by credit card at the time of the interview. Otherwise, you have to collect a form from the embassy and use it to pay in �63 at a bank.
Several people have posted answers, to similar questions on AB, stating that they've simply lied on the Visa Waver Program