Bike-specific shoes have been around for a long time. Basically, the cleats on your shoes clip into your pedals, allowing you to generate power throughout the pedal motion - not just on the down-stroke. Of course, if you're here, you probably already know that. The problem is that the position of your cleats are fixed, with only a small degree of adjustability.

A couple of years ago I read about some people toying around with a midfoot cleat position. Specifically Joe Friel, writer of the Cyclist's training Bible and Triathlete's training Bible, had a really good argument for midfoot cleats which made sense to me.

The idea is that moving your cleat to the middle of the foot effectively takes a lot of the strain off of your calf muscles, as your are decreasing the moment arm between your ankle and the pedal. This can potentially have a large impact on calf fatigue, which comes in play on particularly long days in the saddle, or especially during multisport (duathlon and triathlon) when you have to get off the bike and RUN a great distance.

Since I made the switch, I've done a 2000+ mi bicycle tour, and a very challenging Ironman race. Which I don't have hard data, in my opinion the new cleat position did improve my performance. It definitely didn't have any real drawbacks for me.

The only potential negatives I can see are:

If you are a sprinter and put down some short distance, high power efforts, I would imagine your peak power would be a bit lower with this. This is just a guess, though.
Changing to midfoot cleats effectively pushes your feet forward, meaning you are at a greater risk of toe overlap/ With toe overlap, your toe enters front wheel territory; if you are climbing at a very slow speed for instance, and you turn the front wheel sharply, you run the risk of your front tire bumping your toe. It's happened to me a few times, but you adjust to it quickly and I never considered it to be a big problem.
Of course, by modifying your cleats, you are voiding any warranty from the manufacturer...but that shouldn't surprise anyone here.

Other considerations:

Bike fit will be affected. You will have to lower your seat post significantly from what was previously a good fit. You may also want to lower your stem height (I didn't). So, if you already had a perfectly dialed-in bike fit, be prepared to have to refine it. Personally, I am always tweaking my bike fit as my flexibility fluctuates, so this didn't bother me. There are plenty of instructables dealing with bike fit, so I won't go into that in this post.
There are now more options out there for midfoot/midsole cleat positions offered by cycling shoe manufacturers, so if you are worried about damaging your expensive cleats, this may be a more desirable option.

Step 1: Shoe/Cleat Recommendations, Materials

This method will work best with shoes with flat, plastic soles. Carbon is OK, but you'll need to be careful about drilling. Shoes with non-flat bottoms are tricky. I'd suggest trying to fill in the holes to try and create a nice flat surface to work with. I have found a couple of models from Sidi have a nice flat bottom (mine are Sidi T2)

For cleats/pedals, I really like Egg Beaters. Dead simple, easy to use, and they allow for a lof of twisting (good on the knees for long distances). That being said, different cleats should work with this method, but becomes more difficult. With 3-hole cleats (like LOOK), for example, you will have a 3rd attachment point to plan for, and you'll want to make sure the cleat makes good contact with the shoe. Just something to consider.

As far as tools, all you need is a drill, tin snips, and whatever tool to drive in your cleat bolts.

Additionally, you will need 4 Tee Nuts (T nuts) and matching bolts. 1/4-20 size is very common and can be found anywhere. However, I found that with 1/4-20 size hardware, the bolts tend to loosen over time. Instead, I recommend looking for M5x0.8mm Tee Nuts. Harder to find, but will fit the cleat bolts you already own, plus the tighter pitch on the threads means that they are less likely to loosen over time. McMaster Carr has them, but only in pkgs of 100. I found some on eBay. Optionally, thread locker (such as Blue Loctite) can be used to help secure your bolts.

Step 2: Drill!

I measured the longest ditance on the sole from front to back and marked the middle position on the sole with a Sharpie. Then I eye-balled a horizontal line going across the sole from that midfoot position.

Then, using the cleat as a template, I marked my drill points along that line. I wanted to get close to the instep, but far enough away that the shoe didn't lose any strength.

When drilling, remember that you are drilling through plastic. Drill bits tend to bite down really hard and pull quickly into the plastic. Go slow. One trick I like to use it to drill a small pilot holes, then run the final bit spinning in reverse. I find that this will often make a smoother hole.

Step 3: Snip Your Tee Nuts

As-is, the Tee nuts won't fit into the shoe without interfering with each other. You can snip them pretty easily with a good set of tin snips.

You can probably get away with skipping this step. You'll have to bend a couple of the prongs on the Tee nuts, and they will overlap a bit after tightened, but it might not be noticeable after your insole is back in.

After your Tee nuts are cut, put them in place inside the shoe, and test fit your bolts from the outside. Tighten up the bolts (without the cleats) to properly seat your Tee nuts.

Step 4: Fit Your Cleats!

Now install your cleats in their new position! You can see in my picture that I adjusted my cleat position inward as much as possible. Notice where the cleat is in relation to where the standard mount is. I also used a drop of Blue Loctite thread locker on each bolt.

Step 5: Optional: Add Some Cleat Protection

Now, your cleat is going to be somewhat exposed when you are walking off the bike, so you might want to think about protecting it somehow. For a long time, I simply used some old cleat bolts in the original mounting position, and that was it!

More recently, I designed a simple 3D printed solution using a nylon material that's pretty tough. (.stl file attached)

Now go ride!