So Harry is going to have a lot of time over the summer, one of his primary goals should be to work out more about magic. Maybe we can do some brainstorming some of the experiments he could foreseeably perform given his current resources ch118

Some mostly spoiler free examples I can come up with:

Transformation:

• Confirmation of violation of conservation energy
  • Lift stuff up by increasing mass of weight and confirm that at least in the local area energy is gained even after the transformation is let go.
• Mechanics of transformation
  • Get a wizard born who has no idea of the workings of a computer (they would have existed in 1992) and tell him to transform something into a CPU. Observe what is produced.
  • Pick a book at random and try to duplicate it with transformation
  • A bit difficult for Harry but with magic anything is possibly. Try to transform an object supercooled to the point of being no longer solid but in a different phase of matter into something else
  • More attempts to transform air.
• Mechanics of Time Turners
  • What counts as a transfer of information for the purposes of determining the maximum information flow with time turners. Depending on the specifics it may imply that the restrictions are imposed by the creators of the devices rather than being some constant of nature. 6 hours is a very specific time frame
  • Attempt the various easy methods to break the 6 hours limit. Get someone to apparate you to a different time zone and see if it changes anything once again if it did this would imply that the current limit is an intentional limit rather than being a fact of the physical world.
• Spell creation
  • This is more reading than anything but get access to the restricted books by any means necessary and work out what's needed to create a new spell.
  • Look at the history of new spells. How often are new ones discovered. How does the result and language of new spells change over time. Is it region specific? The results of this may imply that spells are actually being created or that they are just being reinvented or found.
• Mechanics of potions
  • Why are some potions so much more complicated than others?
  • If there is some kind of conservation of energy are the exact ingredients called for needed or will anything with the same energy work?
  • Get a potion to an actual scientist or create the necessary equipment and get some idea of the resulting chemical composition. Try to work out if it's obeying the standard rules of chemistry.

Also worth mentioning that Harry is surely not the first one to attempt some level of understanding of magic. Harry needs to try undertake a systematic analysis of as much literature as exist on magic. What is the faculty of Hogwarts even doing over the summer?