Don't you just love a nice big cup of coffee? I mean a BIG cup of coffee? Me too. The problem is my cup is so big the coffee gets cold before I get to the bottom. Here's a solution.
A cup cozy.

Step 1: Tools and Materials Needed

Tools

Scissors
Sewing Machine
Stapler

Materials

Really big coffee cup
Staples
Thread
½" Velcro® (or other hook & pile fastener)
Spandex bonded neoprene rubber

The material I used was Spandex bonded to both sides of a sheet of Neoprene. It's the same stuff wetsuits are made of.

Step 2: Cut the Bottom

Place your cup on the material and trace the perimeter of the base with a marking pen. Leave some space around the edge because you're going to need that material.
After tracing the perimeter of the base, cut around your marking leaving about ½" of space around the edge.

Step 3: Cut the Side Wrapper

Lay the cup along a straight edge of the material and mark a line down the length of it. Use a straight edge to keep it neat.
Next cut the material, again leaving a ½" buffer of extra material.

Step 4: Notch the Fabric for the Handle

Take your cut strip of fabric and align the marked line with the bottom edge of the cup.
Mark and cut a notch to allow the fabric to slide under the handle.

Step 5: Cut the Fabric for Length

Now that you've notched the fabric for the handle, wrap the fabric around the cup and cut it so it overlaps by about an inch.
Notch the other side of the fabric for the handle.

Step 6: Burn the Edges

The beauty of this material is that it's rubber and it melts. This is an advantage. Instead of hemming, we can just melt it to seal the edges. Be careful because if you hold the flame there too long, it could catch fire. That wouldn't be good.

Step 7: Attach the Velcro

By now we have created the side wrapping of the Cup Cozy. It should overlap itself under the handle and seal itself with the Velcro.
Cut the Velcro to the length of the area under the handle and sew it on to the material.
Points to remember.

We sew this thing together from the inside, then turn it inside out.
The Velcro is attached to both sides of the fabric. The hook is on one side. The pile is on the opposite side and at the other end of the fabric.

Step 8: Sew the Base

The two flaps at the bottom from the previous step still need to be sewn together.
Wrap the fabric around the material and mark it. Sew the material together at the mark then trim the excess.

Step 9: Attach the Bottom

Conventional sewing wisdom dictates the use of straight pins to attach two pieces of material together before sewing. I didn't have any, so I used staples. Staples are way more manly anyway. If you did this right, the perimeter of the side will be roughly the same as the size of the base.
Attach the two together. Sew along the lines that you previously marked.
After sewing, trim off the excess material.