Take the necessary precautions when mixing miticides, i.e. gloves, respirator, long sleeves, long pants as well as covering your head. Abamectin is extremely hazardous. As far as adding anything your friend is using the dish soap as an adjuvent. This reduces the surface tension of the water and is used as a spreader/sticker. Make sure if your using dish soap that its dish soap and not detergent. I recommend Dr. Bronners castile soap.
The use of an adjuvant when dealing with mites is very important. It increases the effectiveness of the miticide considerably. In addition to using an adjuvant I also highly recommend you set the pH of the final mix between 5.5 and 4.5. Pesticides are more effective when the pH is in this range.
The best adjuvant I've found is Indicate 5. Its a spreader/sticker that automatically sets the pH for you. Here's a link.
http://rosemania.com/shopsite_sc/store/html/product40.html
As far as mixing use the best water you can get whether its RO or distilled. Add in your abamectin followed by your adjuvant. If you're using Indicate 5 it will turn pink when the pH is is correct.
Hope this was of some benefit to you.