Water sensing circuit?
I was learning about GFCI and wanted to know if there was any easy way to create a simple circuit to detect when there is water.

Yes, detecting water is quite possible, but continuously detecting water in such a way as to know what circuits it is affecting would be complicated.  The GFCI circuit requires access to both the full input and output current to detect imbalances.  In theory you could provide this function on every voltage converter, but it would increase complexity.  That circuit wouldn't detect water, it just detects shorts to ground.

What I would like to do is make a LED turn on if a component (whether it's a wire or anything else... however, not any expensive water sensors) in the circuit gets in contact with the water.

Rather than doing that, you should probably just shut the circuit down the instant a ground fault is detected, possibly with a report or indicator, but the shutdown is the important part, or why bother sensing?  The difficulty will lie in accurately comparing the output and return currents so as not to nuisance trip or fail to trip.

Here's what I'm imagining: if water leaks from somewhere, maybe some part or component of the circuit gets wet and fails (or one pathway becomes inactive), and so the current takes a different path, which leads to a LED turning on and staying on until it's manually reset.
Is there a simple way to do this? Thanks a lot!

A simple way would be to provide some protection per power supply or per board or per voltage.  GFCI protects faults to ground though, not open circuits, so it would be attempting to save the circuit by shutting it down before it had a chance to run for long under fault conditions.  For circuits with voltages that present danger to humans, GFCIs are also a valuable life safety apparatus, but at small voltages in consumer electronics that's less relevant.
Another thing to note is that if you think your project will get wet often enough that it needs an indicator to show you that it has been wet, you may wish to consider sealing it as a priority and use more elaborate methods only as backup.