If your hair is brown now, you're going to need to bleach it to really get any bright colors to stand out. If you're not into the idea of this being permanent (or requiring re-dying to get rid of it) then don't waste your cash on dye that won't show up well.
As another concern, just dying parts of your hair something like green and leaving the rest brown can look a little shabby. So if you're going to do super out-there hair, do it well. If you don't want a whole headful of neon hair, dye some sections the color of your choice and leave the rest black or blonde. This way you'll have a cohesive hairstyle, but you won't be over the top.
What color you choose depends on what colors you wear most and what colors actually look good on you. It stands to reason that if you don't look good in a blue t-shirt, you're probably not meant for a headfull of blue hair. If you find yourself always wearing red outfits, it's probably not a good idea to dye your hair green unless you like the whole Christmas look.
Colors like red, blue and green are always good standbys for guys. Shades of neon pink and purple can look just as good as long as you have the overall look to pull it off.
Been dying odd colors for years.