Although it may seem easy enough, saying hi to a girl can be one of the most difficult steps you take to overcome shyness. Despite the perceived or genuine implications, such as rejection, taking steps to reduce your feelings of anxiety when approaching someone can also help reduce the extent of your shyness. Over time and with positive interactions, you might get more comfortable saying hi to other people.
Surround Yourself with Support
Bring along a friend or two when you approach a girl. Surrounding yourself with social support can help decrease your anxiety and increase your self-confidence. Friends provide emotional support and, ideally, are not going to make you feel worse if your introduction isn't well received. Prior to approaching the girl, speak frankly with your friends about your intentions as well as your hesitation in making this move. Your friends might give you tips based on how well they know you, and they can also be relied on for help escaping if you are feeling particularly anxious or vulnerable when speaking to the girl.
Ask a Question
Approaching a girl with the sole intention of saying hi may not provide you with enough to continue a conversation once the ice has been broken. This can lead to awkward silence and increased anxiety for one or both of you and is not conducive to continued discussion. Instead of just saying "hi," have a question or two in mind before you approach. Keep the topic light and relevant, such as the weather or mutual interests, and steer clear of emotionally charged, polarizing topics like politics and sports.
Make Her Laugh
Tell a joke. If you're feeling shy when approaching a girl, chances are that she is also feeling a bit of anxiety. Telling a lighthearted joke can break the ice and lower the walls that anxiety can create. Be sure the joke is easily understood and at least mildly funny, to prevent feeling awkward and at a loss for words. Test out jokes with friends and family prior to using them on the girl. Make sure the joke isn't terribly long, which can lead to the loss of the girl's attention. Don't use children's jokes that everyone has already heard; the last thing you want is the girl beating you to the punchline.
Distract Yourself from Shyness
Carry something with you to help you focus and remain calm. Putting a stone, coin or similar object in your pocket can help you focus on your delivery and distract your thoughts from anxiety or shyness. Hold and handle the object with discretion; don't become too distracted by it to speak. Keep eye contact in mind when speaking with the girl, and imagine your energy and strength being generated from the object in your pocket.