I doubt the trial would halt if anyone in Jodi's family was ill or had passed away. Unless it was Jodi herself, of course. Also, she would be considered too high risk to be allowed leave to visit an ill family member or to attend a funeral service. Her family are spectators in the courtroom and should they not show up for whatever reason, court would happen without them.
I think the emergency involves a close family member of a primary player...could be anyone from the judge to a juror, any of the attorneys, or the current witness. Or the killer is claiming to be sick.
I have been hoping this delay was because the state had decided to take the DP off the table and allow this atrocity of a trial to, for all intents and purposes, finally draw to a close. But many tweets have said it has nothing to do with the actual trial so I don't think that's it.