I was raising a garu/durahan and he was your average monster
in adventures. He would often find toy glasses, teddy bears, and
other useless junk. Then one day he learned Lucky Star. Every
single item he found by searching or in a chest from that point
on was either a medal, a document, a precious stone, a
combination material (including a lucky star) or a Training
Gadget thing. Even in a normal red treasure chest, I found a
Lucky Medal! In short, the lucky star is a valuable ability for
those looking for Training gadgets, Medals or any other rare
item.

P.S. If you do not believe me, take a monster with lucky star on
an adventure and have him out whenever you open a treasure
chest. Than switch to another monster. The results will be quite
obvious after opening only 2-3 chests with each monster. (note:
open only red chests as green chests have rare items anyway.)

Wait, problem. Found a Junk Toy in Promiass. I think it highly raises your chance of finding rare items, but it does not guarantee it. Still, it is very useful as it does give you more rare items as 1 common item in three adventures is extremely good. It definitely requires more testing though.

i have a durhan and he learned lucky star but when i go to adventures i never found a good item i always find str stone, tgh stone etc. so i don't think i always brings good fortune.

In my experience, there seems to be a correlation between monster rank and items found. I've found all of my Medals, and most of my combination items, with A- and S-class monsters who had Lucky Star.

I also seem to have better luck if I have more than one monster with Lucky Star.