That's a complex and abstract question. Are we counting as going from Earth's surface or orbit, and if it's an orbit which one? Are we counting the cost of launch infrastructure in this? Does the object need to survive impact? Is there a time restriction on how long this can take?

Overall it's a question of lifting mass. Rockets are reaction engines, you need to expel mass in order to move. However, you also have to lift that mass in the first place, which means you need more fuel to lift your extra fuel. It's a massive feedback loop and increasing the payload (or delta-v requirements) results in exponential growth of the total mission load (and thus cost). Staging is very important - if you drop the early heavy launch stages the later stages need to move a much smaller mass, hence the traditional distinction between rocket and spacecraft. Big rocket gets you near orbit, medium rocket kicks you into orbit, small rockets on the spacecraft kick you onto a translunar trajectory, into lunar orbit, then send you back.

You can really divide the mission into two phases. The first phase is reaching orbit, and it's currently where the vast majority of the fuel is expended. Chemical rockets have a fairly low infrastructure cost, but as a result of the fuel equation they have absolutely insane operating cost. Spaceplanes can be more efficient over their entire lifespan because you're not throwing away the whole airframe with every launch, but they tend to have inefficient staging that becomes exponentially worse as you go farther out (you don't want to have to lift aircraft engines into space, or move your orbital injection airframe into lunar orbit, etc). Everyone likes the concept of Single Stage To Orbit but there's a reason the Shuttle drops its boosters and fuel tank.

If you can find another way to get your craft to space you greatly reduce the overall cost of your mission. Space elevators are the fantasy but require exotic materials like carbon or diamond nanothread. The launch loop is an interesting design that can be attained with present-day materials. Another option would be to manufacture your craft in space. If you pull an asteroid into orbit you have an unfathomably abundant source of materials. An average asteroid has about 10x the quantity of precious metals ever mined in human history as well as craptops of iron, nickel, lead, etc. And it would all be in easy reach, outside the worst of Earth's gravity well. The husk can be used as an Aldrin cycler. These all have fairly large infrastructure costs, ranging from billions to trillions, but would offer very low-cost payloads once built out. For example a launch loop would cost about $30 billion and reduce launch costs to $3/kg. In comparison SpaceX spent about $400 million to develop the Falcon rocket and costs $1k/kg for payload, and this is a comparative bargain for chemical rockets. The first hamburger is very expensive with alternative launch methods, but in the long term the price is much more reasonable.

The other half is the lunar phase of the mission. You need to get the object out of Earth's gravity and into the Moon's. In theory as long as you leave the Earth's sphere of influence and are inside the Moon's it will eventually be pulled in and impact the moon, but this could take a really long time and you are giving up a lot of control over how the payload arrives. If you want a delicate payload to arrive safely you need to allocate weight for motors, fuel, and a guidance platform. Take a look at the mission profiles considered for the Apollo missions - Lunar Orbit Rendezvous (you inject into lunar orbit, then later begin a descent) or Direct Lunar Ascent (you aim straight at the lunar surface and brake yourself in one go). Direct Ascent is faster but takes more fuel, so we went with the LOR profile.

Other missions can have very different needs. For example, for long-term long-distance unmanned missions, electrical/ion rockets become very compelling. They have superior thrust-to-weight but very low thrust levels - so you can accelerate or decelerate a lot (delta-v) but only fairly slowly. For travelling interplanetary/interstellar distances at human-relevant timescales you need a stronger kick. The best design so far is nuclear pulse propulsion (see: "Project Orion"), which operates by chucking nukes out the back and riding the shockwave. It's a childish idea (I came up with it independently myself as a child) but it works - it gets you large payloads (it actually works better as it scales up) moving at significant fractions of the speed of light (0.10-0.33c). It's an environmental disaster, of course, you couldn't fire the engine anywhere near a planet.

Intelligent use of orbital mechanics can further decrease your needs. For example you can "slingshot" through a gravity well, which lets you steal an infinitesimal amount of the planet's velocity. If you burn your engines as you descend, you keep your velocity but also don't have to lift the mass back up again. However, the planets rarely align that this is a useful maneuver. This is actually a general rule - getting there quickly takes a lot of fuel and if you are willing to wait for orbital mechanics to work in your favor you can greatly reduce fuel requirements (see: Hohmann Transfer). The extreme is the Aldrin Cycler - a spacecraft placed on a periodic orbit between Earth and Mars with a duration of several years. Once injected into its orbit would require no fuel except for small corrections.

Specific response to your question - the cheapest one-off is a small, staged rocket that pushes itself just over the boundary into the Moon's sphere of influence and then gradually arrives however many years later. If you specify that it needs to be verifiable, you attach a proximity-fuzed grenade that scatters a cloud of retroreflective material onto the surface, which you can identify using a laser. That frees you from the need for guidance or deceleration to prevent a catastrophic impact. You do need to give it enough of a trajectory that assures it arrives on the light side of the moon, however. Time limits are handled by increasing fuel load.