In an upcoming campaign I’m playing a Lizardfolk barbarian, and one of the lizardfolk features is Cunning Artisan which means I can harvest any creature, plant, etc to create a club, javelin, blowgun darts, or shield from what I harvest. So my question is would the stats for a club change depending on the material? I looked it up and saw that the base die to roll for damage is 1d4 for a club, so if a wooden club has 1d4 of damage then would a bone club do more? What about a dragonbone club? I assume it would, but I can’t find an answer anywhere.
A club is a club.
There are no rules in the game for different basic materials or styles of construction providing different weapon stats.
The only special material effects in the rules are Silvered weapons (Player's Handbook p.148) and Adamantine weapons (Xanathar's Guide to Everything p.78), which primarily overcome certain kinds of damage resistance (and adamantine weapons do extra damage to objects). A few specific magic items in the Dungeon Master's Guide call out particular materials, but that's more in the way of description rather than ascribing specific powers to particular materials.
If the ability says you can make a club, then you make a club, nothing more and nothing less. The main benefit of this is that you can quickly craft actual weapons, rather than using something vaguely club-like as an improvised weapon, which is what it would be if anyone else tried to use parts of defeated enemies for the purpose. Normally, crafting requires downtime, so you can't even do it in the field; converting a branch or bone into a club isn't something the average character can do during an adventure, but you can do it while you're taking a short rest.
If for some reason the creature you were disassembling for parts had significant silver or adamantine in them (perhaps some kind of construct), the DM might be inclined to count the crafted weapon as having that material quality, but that would be strictly by DM fiat and not because the rules said so.
No, the stats are the same.
There are no rules that change the stats of a weapon if it is made from unconventional material or construction. The only RAW that addresses "similar weapons" are found in the PHB under Equipment - Weapons under Improvised Weapons:
An improvised weapon includes any object you can wield in one or two hands, such as broken glass, a table leg, a frying pan, a wagon wheel, or a dead goblin. Often, an improvised weapon is similar to an actual weapon and can be treated as such.
So a club is a club, regardless if it's a bone club or metal club, the stats are the same. Looking at the list of Weapon Properties (Finesse, Heavy, etc.), different materials aren't listed and therefore this doesn't affect the stats themselves. Based on the weight of the materials you use for your club and other weapons, a DM can use the rules for Heavy or Light weapons even if the size remains the same (e.g. a lead club would be much harder to handle than a wooden club):
Heavy: Small creatures have disadvantage on attack rolls with heavy weapons. A heavy weapon's size and bulk make it too large for a Small creature to use effectively.
Light: A light weapon is small and easy to handle, making it ideal for use when fighting with two weapons.
The only material specifically called out in the PHB is for Silvered Weapons, which allow players to damage certain creatures that have immunity or resistance to nonmagical weapons:
Some monsters that have immunity or resistance to nonmagical weapons are susceptible to silver weapons, so cautious adventurers invest extra coin to plate their weapons with silver.
The other material mentioned in the RAW is Adamantine in XGtE:
Melee weapons and ammunition made of or coated with adamantine are unusually effective when used to break objects. Whenever an adamantine weapon or piece of ammunition hits an object, the hit is a critical hit.
In terms of using said weapon, the PHB states:
At the DM's option, a character proficient with a weapon can use a similar object as if it were that weapon and use his or her proficiency bonus.
This is more of a question for your DM. RAW there is no difference in damage and I don't see how a bone club could do more damage than a wooden one, unless you add spikes, but that would probably just change the damage type.
I myself would consider dragon bone to just be a more durable material than wood, but it all depends on what your DM allows you not looking at RAW, considering its rarity and that it's infused with natural magic.