To synthesize you must have a weapon, preferably a crap or cheap one and a item such as a white potion (your going to need 10 of these but if you are also going for synthesize equipment 20 times then you will need 20) and a lot of money. Each time you synthesize you will need to pay money. Save before doing this so you can reload if you want your money back. You can synthesize at any idol. Basically sinthesize the weapon 10 times and the achievement will unlock.
You should get lots of white potions so dont worry about that but if you need money then i suggest playing on extreme difficulty as the enemies drop alot more money than on easy.