I don't know If someone did this before, but I didn't see it, so, this is a guide for the fossil mission, cause is the worst thing in the game:
Use the shovel to find the mysterious fossils (best thing here would be to have like 3 or 4, and go to the museum, and use cheat engine putting the actual number you have, and after giving one to the girl, change the number on cheat engine and repeat until you have the item on cheat engine, and then change the number to 99 and you'll have infinite fossils doing it every time you want. If you don't use cheat engine or other programs to hack it, you'll be there for long...)
This is the list of all fossils you'll need to have (I recommend you to write them to know what fossil you're missing so you don't have to waste time checking if you already gave her one fossil):
Amber
Ammonite
Ancient Art
Arandaspis Prionotolepis
Compsognathus
Coprolite
Fossilized Fern
Fossilized Fish
Gigant Nautilus Shell
Isoletus Rex
Mammoth Tooth
Nautiloid
Smilodon Tooth
Trilobite
Velociraptor
Carcharodontosaurus Skull
Mosasaurus Skull
Tyrannosaurus Skull
It's a little crazy doing this without knowing all the names, so there you have, I hope it helps, and also would be great if they decide to change this mission so you can't get the same fossil once you get it