Kids bike rear derailleur change
An upgrade to the drivetrain on this bike is very likely not possible.

A 7 speed drivetrain either has a old-style freewheel, where the sprocket cluster and ratchet mechanism are in a single unit that threads on the hub. These can't be upgraded to more sprockets.

The hub may have a newer style freehub design where cassette sprockets slide on a splined cylinder, but 7 speed freehubs are narrower than those designed for 8 speeds or more. 

The wheel might have a 8+ speed freehub and a 7 speed cassette installed with a spacer, but this is unlikely.

When answering questions about upgrading the drivetrain, most of the time my advice is that's it's not worth the cost. Separate drivetrain components are expensive, changes in one place have a knock-on affect that require other components to be replaced. Sometimes the entire drivetrain including shifters, crank and rear wheel need to be replaced (although not in your case). Also, you don't get the full benefit of better quality components when they are installed on a lower quality frame.

Personally, as this is a kids 24" wheel bike I'd leave it alone and save the money for your daughter's next bike.