The cheapest way is to do it "standard" service rather than waiting till the last minute and paying extra for expedited service. There is a rule change when a child turns 14--essentially, a 14 year old can apply for his/her own passport, under 14 needs both parents or one parent plus documentation why the second isn't there. Under 16 is a child, good for 5 years, a little cheaper. 16 and up is an adult, good for 10 years.
The only reason that I can see for waiting is if one of those "transition" times occurs in your family. For instance, I will wait till DD turns 14 rather than go thru the hassle of explaining how/why she has no father. Waiting till someone turns 16 would cost a little more, but gain 5 additional years of use.
All said, we're going to Europe in a year--we'll need em. What a summer--Europe and Panama cruise!