How to inherit the default methods when writing your own custom methods?

The issue is not that your methods are interfering with Roblox's. You are actually creating a table that has a part at index part (in Lua .part is the same as ["part"]). When you call the :Destroy() method on your variable "part" you are really calling the :Destroy() method on your newContainer table that you return from the Container.new function. You would need to call the :Destroy method on the actual part, which you have at index part in the newContainer table. Here is what your fixed code looks like, with a wait added in to show you that the part is created and then destroyed.

local Container = {}
Container.__index = Container

function Container.new ()
    local newContainer = {}
    newContainer.part = Instance.new("Part")
    newContainer.part.Anchored = true
    newContainer.part.Parent = workspace

    setmetatable(newContainer, Container)
    return newContainer
end

function Container:PrintName ()
    print(self.part.Name)
end

local part = Container.new()
part:PrintName()
wait(3)
part.part:Destroy()

You even do this with self.part.Name in the PrintName method, I just think you missed what the part variable really is (a table). I recommend that you name your variables something that helps you remember what is actually inside of them, and don't confuse you like this.

I hope this helps. Have a great day scripting!