I would like to attach a rectangular piece of wood (without drilling through the wood) to a string to make a necklace. The piece of wood should be attached with a knot to its extremity, and hang vertically. I would like the knot to be small, easy and aesthetic. I am thinking about using a fisherman's knot to make a loop out of the necklace, and then somehow attach the wood to the loop - but it might be more clever to attach the wood to the two ends of the string.
You may have luck with hitches or whippings. The former tends to involve attaching a rope to something, such as a piece of wood. The latter finishes the end of a rope to keep it from fraying, or finishes a piece of wood...to keep it from splintering? Both tend to have styles that can be repeated down a bit of wood making them both more secure and more attractive.
The specific knot isn't critical as long as the knot has the functionality. There are endless choices, so you can pretty much pick one that you find attractive as long as it is a type that doesn't slip (and any knot reference will describe that characteristic). However, you will need to deal with the issue of any knot attached as described in the question pulling off the wood; just a single knot of any kind won't work because there's nothing to keep it from sliding off. So rather than suggest a specific knot, let me focus on securing the wood to the string regardless of the knot you choose.
If you're amenable to a dot of glue on the backside, you can hang it from one corner or one edge of the wood (the orientation wasn't clear in the question). Create a loop in the string sized to fit the desired distance from the corner of the wood or around the sides if hanging from one edge. Hold the string in place temporarily and apply a small amount of glue between the string and the back of the wood.
The string can hold the wood securely if you capture two diagonally opposite corners, or all four corners, each in a loop, where the diagonally opposite loops are on the same run of string and can't move apart. For the wood to lay flat, the string that goes around your neck should come either from the top corner, or the two corners of the top edge, depending on the desired orientation.
If you're amenable to using the edge of the wood, you can do it with the method used for rimless glasses. There are several approaches. One is to file a "V" groove around the edge of the wood using a tiny triangular file to a depth of at least 1/3 the diameter of the string. Then loop the string around the groove, stretch it tight, and secure it with a knot. That will hold up to normal jewelry handling. But if the string stretches a little over time or the wood gets a good tug, it could pop out. If you want it permanently secure, you can use a small amount of glue in the groove.
A variation on this is far less visible than a border of string. Use either clear nylon fishing line or nylon or polyester thread, or a strand of wire. These are much narrower but strong. The groove can be much narrower, and the strand can be totally sunken into the edge. Nylon will virtually disappear. Wire will hardly be visible, but you can pick a wire color to make the thin line an accent color. Form a loop at the hanging point and put the string through it. You can always add decorative knots if you want it more elaborate.
If you want the wood to hang in the rectangular orientation rather than the diagonal orientation, you could just glue the string to the top edge of the wood. Then you can add decorative knots if desired.
If you don't want to glue the string on, you can secure it with a tiny "staple" near each corner. Bend a fine wire into a rounded staple shape the diameter of the string (if the string is stranded, you can even make the staple smaller and capture just a strand or two of the string). Pre-drill the nearly microscopic holes in the top edge the same diameter as the strand of wire (micro drill bits are sold in sets going down to a small fraction of a millimeter in diameter for making such tiny holes, and they aren't very expensive).
Capture the string and then use needle nose pliers to push the wire into the holes. They will typically be secure from friction. If they are too loose, coat the tips of the legs with a tiny amount of glue before pushing them in, and hold or clamp them in place until the glue sets (which shouldn't be long for such a small amount).
You stipulate no drilling through the wood. It isn't clear whether your objective there is to avoid a visible hole through the face of the wood or not wanting any damage to the wood. If you just don't want to mar the face, you can make a tiny hole on the edge and insert an eye pin (glue-in or threaded). That gives you a loop for the string.
The glue-in type uses a tiny hole (use the same type of drill bits described for the "staples"). The eye pins are available at hobby shops and jewelry supply stores. You can even make your own glue-in type by bending a piece of wire. The glue-in type are typically pretty stiff, so you may be able to make the hole the same diameter as the wire or a smidgen smaller, then push the pin in with needle nose pliers. That will usually hold well without glue. For the screw-in type, you just want a pilot hole the diameter of the shank under the thread. The jewelry-sized threaded eye pins are pretty small, so the pilot hole may also require micro drill bits.