You'll need to hunt down monsters on tier 3 planets (I believe that's Ocean, Savannah and Snow planets) while using an Avikan hunting weapon. These weapons include the Hunter's Knife you start out with, as well as any Avikan bow, hunting spear, or throwable weapon. You can find higher quality bone materials on higher difficulty planets, just like how you can find better ores on more difficult planets.

Thanks!

I hate to necro this, but does the Keffrider's Spear reward from the Avikan questline count as an Avikan hunting weapon? I know it counts as a regular hunting weapon, and I was getting Monster Plates and Monster Bones with it on a Mountainous planet (rating 3.3 on the weapon loots I saw), but I did not get any Monster Claws on probably 100 or more kills. Thanks.

I must have not been on a high enough level world. I went to a Fauna threat 2.9 world and started getting Monster Claws with the Keffrider spear right away.