You have quite a bit of play room here, considering the scale length is much shorter than a standard guitar. You could tune-up the strings, or slacken them even more; but be aware of the limitations of the stock strings as they will have a point in which they begin to sound "not good" due to the lack of tension on the string.
However, I'm wondering if the strings were properly installed:
I managed to overtension one of the strings and it sprung, uncoiling a bit just at the tuning machine. Clearly my fault, I gave her the money back and I'll get another string for it and tune it properly.
The "uncoiling a bit just at the tuning machine" is almost always a tell-tale sign of improper installation of a string.
If you want to get nerdy about it, D'Addario makes a great site called stringtensionpro.com which could help you out. Disclaimer: it is a solicitation for D'Addario strings but the specification of the strings can be matched identically for your purposes with a bit of measurement.
This tool can show you something like this nylon string chart I just created:
In it you see a "reference set".
The left-most chart shows you normal tensions on a normal set tuned to E standard (An extremely common 6-string guitar tuning). The tensions of each string hover between 11 and 16 lbs of pull for each one.
With a 16" scale length that tension is drastically reduced to about 4 or 6 lbs of pull on each string.
Adjusting the scale length of a typical set of strings will easily allow you to see what the effect is on a 24" scale length. However I can tell you that's only .75" less than a standard Gibson scale length, it's much closer to a standard-sized guitar.
With that said it looks like E standard tuning (EADGBE) is safe on a small guitar like you have; but ultimately the bracing on the soundboard of the guitar will determine how much tension it will allow. Keep in mind the audience as well; these are for small children. The number one concern for children learning guitar is the pain experienced by pressing down the strings. The lower the tension on the strings, the more "comfortable" it is to play with the child develops callouses. We're already dealing with less than half of the normal tension of a guitar; that may be just enough. However we're also making the child use steel strings; so the pain factor may be two-fold.
If the child is truly a beginner they may also benefit from an open tuning. An open tuning is simply a tuning where the strings are tuned to a more pleasant sounding chord. Slide guitarists do this all the time. A personal favorite is Open D (DADF#AD), but you could easily do any variation of a major or minor chord to great success. The best part about this is, for a beginner - who more than likely will not be taking lessons (I only say this as guitar instructors generally don't recommend toy guitars for beginners) - it allows music to be made immediately as placing fingers across the strings and strumming automatically creates somewhat musically pleasing sounds. A major benefit to most open tunings (like Open D) is that the string tension is loosened a bit, alleviating finger pain associated with more traditional tunings.
At this stage - with a toy guitar - it's more important to get the interest in playing music for the child. Slacking the strings a bit and/or using the open tunings will prevent the strings from coming undone, and allow for "music" instead of "noise".
Edit: Adjustments made as I thought this guitar was string with nylon strings, all the same rules and relativity still apply.