I posted a 3x3 sliding puzzle game made with Lua on the wiki under script index page.
Other puzzles you could do with Lua or potentially with action parts are...
ring/tile rotation (rotate tiles/rings to align the pieces until the correct image/pattern is shown).
chess / checkers (potentially)
pipe puzzle (essentially create a path from a starting point to an end point).
match 3 - or any other kind of match game you can think of
connection - I've forgotten the name for this, but it's where you draw a line between 1 thing to another thing & connect them together, like this...
physics/collision based mini games (possible)
text input (password)
qte (quick time event) - maybe have to spam mouse button or click certain spots on the scene within x time limit
combination lock (keycode) - tumbler lock or digital keypad.
There are so many different puzzles you can do with or without Lua script. I'm sure I've probably forgotten a bunch of stuff too.
Speaking of puzzles, it might be a good idea to create a wiki page about all the different types of puzzles someone could potentially include in a point & click adventure game as there are many more than what I listed.