That brown spot in the white part of your eye is known as 'primary conjunctival melanosis' which is a collection of melanocytes, similar to what forms a mole on your skin.
Most of the time it is harmless, except for its disturbing appearance. We like to follow up on these 'moles' to see if they are growing in size, because very rarely, they may turn cancerous (only if they are growing). If there is any suspicion that it could be cancerous or if it is very troubling cosmetically, we can remove it by a simple excision. Otherwise, you may leave it alone and just keep a watch on whether its growing.
Hope that helps!
Dr Claudine Pang