To be honest I would not make such a big deal out of it, Brasero. This was your first date with the girl? You cannot expect her to ditch her family for a guy she just met. If I had a family dinner, I would probably cancel on the girl as well, even if I was initially interested in her.
Go with detox75's strategy. Tell her you're busy tomorrow and schedule another time, but please do not let this be a dealbreaker. It shows she cares about her family as well and does not immediately run to any random guy she just met. Sure, you might not have her at the level of interest you want her to be, but this usually takes time with girls. Just a bit lame she cancelled on such a short notice in my opinion.
Schedule a new date and have such an amazing time with her that she becomes so interested in you that she'll not cancel a second time
