Why are you overthinking this?

I'm not saying that the girl did this for no reason, because I certainly do not believe that. Even that young, children aren't irrational. Their logic is sometimes dificult for an adult to follow, but it's there.

The girl wanted you to hold onto the cap for a few minutes (or seconds). It's possible she thought you would need it. It's possible she wanted to see how you would hold it. Whatever the reason, it really doesn't matter.

As for the trust possibility, that's possible, but definately not for the reason the person suggested. The girl may have felt (perhaps simply out of gut instinct) tjat she could trust you, and wanted to express that feeling of trust. Considering how often we adults fail, with all our linguistic capabilities to fully express our trust in one another, a gesture like the one this girl used may well have been speaking volumes. Assuming this is the case, it isn't about her needing someone she could trust to hold the cap, it's that she offered the cap to show that she trusted you.

For me to be more specific, I'd have had to have been there.

Basicly though, just accept that you had a special moment, even if you aren't able to fully understand it. Mysterious as it was, the girl was giving you attention, and you should enjoy that fact.