http://www.dol.wa.gov/forms/420040.html
You need the kid to sign it, at a bank, or wherever you want (has to be notarized)
Quoted from the site:
To replace a title:
Complete an Affidavit of Loss form. All registered owners must sign the form, and the signatures must be notarized. If there is a lien holder on the title, the lien holder must apply for the duplicate title.
Submit the completed form:
by mail with a check or money order for the fees to any vehicle licensing office.
in person at any vehicle licensing office with cash or a check for the fees.
Your duplicate title will be mailed to you in 8-10 weeks.