When you lighten dark hair, you run the risk of really damaging it in the process. The cuticle of the hair becomes more porous and it can't hold colour as well, including any colour your stylist put on your hair after highlighting it. Brunettes usually have serious problems with their highlights turning brassy because there are more red undertones in their hair naturally.
Use the colour wheel to help you. Find the colour of your hair (usually yellow or orange) and look to the colour on the opposite side of the wheel. This is the base colour that you should use to fix your highlights.
Try using a colour enhancing shampoo or conditioner with a purple or blue base. You can make your own conditioner by combining equal parts of regular conditioner with a purple or blue semi-permanent dye. Use this on a regular basis to keep your highlights from turning brassy (the sun can also have this effect).
Put a toner on your hair periodically. This is like a transparent coating that tints the hair to your desired colour.
Apply protein filler to your hair to help it become less porous and hold colour. You can dye your hair the proper colour over this filler or you can simply use coloured filler. This is available at a beauty supply store.
Wash your hair on a less frequent basis than you normally would. This will help the colour stay true.
Avoid using heat tools on your hair as this can damage it further. This includes a hair dryer, hair straightener, curling iron and curlers. Avoid golden tones; if you want to use a gold colour, stick with a more neutral gold. An extreme gold can make the hair brassy.