How to move part to mouse position?

So before we get into how to solve your problem, let's just quickly go over what you have here, right?

• You can't get the Local Player from a Server Script

• You can't get the Mouse from a Server Script

• You tried to move the part by changing it's Velocity

• mouse.Hit.p is a Vector3 value, so you don't need to put it inside of another Vector3 value

How to fix this

local player = game.Players.LocalPlayer
local mouse = player:GetMouse()

local part = Instance.new("Part") -- You can just change this to part.Parent like you were doing before.
	part.Parent = workspace
		part.Anchored = true
			mouse.TargetFilter = part -- The mouse will ignore the part so it doesn't end up above the mouse position

mouse.Button1Down:connect(function()
	part.Position = mouse.Hit.p -- mouse.Hit.p is a Vector3 already
	-- mouse.HIt.p is equivalent to Vector3.new(mouse.Hit.p.X,mouse.Hit.p.Y,mouse.Hit.p.Z)
end)

I'm not sure exactly what you're going for, but if you wanted everyone to be able to see this part, you could fire the server with a remote event on click, and you can pass in the part's position as the argument.

Link on Remote Events/Functions

EDIT

I misunderstood the question apparently, so i'll just leave this here.

local player = game.Players.LocalPlayer
local mouse = player:GetMouse()
local part = script.Parent -- Just assuming there's something here based on your script.
local startPoint = part.CFrame.p -- This will be the firing point basically
local speed = 100

mouse.Button1Down:connect(function()
local lookPoint = mouse.Hit.p
local projectile = Instance.new("Part") -- Creating the projectile
	projectile.Parent = workspace
			projectile.CFrame = CFrame.new(startPoint,lookPoint) -- Projectile starts at "startPoint", while looking at "lookPoint"
			mouse.TargetFilter = projectile -- The mouse will ignore the projectile so it doesn't end up above the mouse position	

local BV = Instance.new("BodyVelocity") -- Creating body velocity
	BV.Parent = projectile -- parenting it to the projectile
		BV.MaxForce = Vector3.new(math.huge,math.huge,math.huge) -- Setting max speed
	BV.Velocity = projectile.CFrame.lookVector * speed -- It will fly in the direction the projectile is facing multiplied by the Speed
end)

The rest of the work is up to you, that means utilizing RemoteEvents.