It is always scheduled the day before Father's day. That's just the way things work out and it meshes with other museum/historical site events in the area which is a consideration (Sully Plantation, pretty much across the street has a big event on Sunday).
This is one of the big events (family days) at the Udvar-Hazy. The only one larger is the Halloween Air and Scare event. Originally when we met with the folks who were fundraising to build the facility and lugging the building model around they were saying you would be able to taxi to the museum as a visitor. That got shut down for a number of practical (liability and security reasons). They then said they could open the gates four times a year (it takes a combination of Smithsonian security, Dulles Airport Operations, ATC, and TSA to get the two gates open between the 1R runup block and the museum ramp). My wife, when still a visiting teacher, spearheaded the education department into grabbing one of those days. By and large it's the only regularly scheduled opening (there have been a couple of helicopter days where they just landed directly at Hazy which didn't entail opening the gate).
Anyhow, there are tons of stuff going on inside from small kids learning about traffic pattens flying around on a runway drawn on the floor to various other events for all ages plus aircraft outside including everything from United's 777 down to Cessna 170's, 1939 Waco cabin biplanes to the latest twin engine Diamond (with two 4 cyl stock Mercedes turbo diesels on it, this one showed up at my memorial day party...very sweet).
One unexpected thing was at the end of the outside portiion where they cleared the ramp of visitors and everybody was taxiing back to Dulles proper, we found many intrepid souls exited the museum and were standing staring in through the taxiway fence at that operation. I guess after 35 years of flying my own plane I got immune to it, but there are still people who want to watch all those engines fire up, if only to taxi a few hundred yards before they disappear into the trees.
By the way, it's like 2 miles from Dulles to the musuem on the taxiway. If you start at the Landmark FBO like I did last year it's a five mile taxi.... here's a video of a perspective most people never get to see...
https://www.youtube.com/watch?v=LdXpfdiawms