In The Great Gilly Hopkins, the narrator says, “This was not the time to start dissolving like Hot Jell-O.” What does that mean?

This is a good example of what's called a simile, a literary device that compares two different things using the words "like" or "as." In this particular case, it's Gilly's emotions that are being compared to hot Jell-O.
Gilly has just turned up at her new foster home. As she's unpacking her things, she looks at an old photo of her mother, Courtney. Courtney is a very beautiful woman, and Gilly wonders why she hasn't inherited her mom's beauty. All of a sudden, Gilly starts getting a queasy feeling in her stomach; this always happens when she looks at her mother's photo. There's something about the word "mother" that triggers such unpleasant feelings.
But Gilly knows that she has to be strong. She's just moved into her new foster home and can't look weak in front of her adoptive family. In other words, she can't go to pieces and start dissolving like hot Jell-O. If you've ever seen hot Jell-O dissolve, you'll know just what she means.