We have been both the last week of March (twice) and (for the first time this year) the last week of June. For our family (teenagers), June was much better for one reason -- park hours. Even though it is the start of the Spring break season, unless Easter is very early, WDW's park hours are still quite short in March, with even MK closing at 7:00 last time we were there then. In contrast, last week there were several EMH nights that went until 2:00 am, and MK always stayed open until at least 11 pm, MGM until 10. Yes, it was hot, and yes, it rained, but those were minor inconveniences to us since we knew we could take nice swim breaks and still have hours to spend in the parks in the evening.
So I guess my advice for your family would lean toward March, but we'll go again in June in a heartbeat.
For the poster who asked about spring break crowds, a lot depends on when Easter falls. The Spring break time frame is generally considered to be about the second week of March through the third week of April, with the two busiest weeks BY FAR being the week before and the week after Easter. Easter is April 8 in 2007, so the first two weeks in April will be packed. In 2008, it will be March 23, so that will push the big crowds well back into March. Hope that helps!