Hi.
There really is no one right way to deal with body hair. Different people do different things. Some get rid of it as soon as it sprouts, while others let it grow out and then get rid of it.
Some people don't ever remove it. Still other people go back and forth--shave their armpits regularly in the summer, for example, but don't touch them once the weather gets cooler. Visible body hair is really nothing to be ashamed of, since everyone has it somewhere at some point in their life. It is all really a matter of personal preference--nothing more.
If you want to suggest something to your friend about her body hair, you might want to be direct and let her know about removal options that you recommend. Also, you could tell her why your own armpit hair bothers you, if you're worried that she might not understand what you are implying.
If you think bringing up her armpit hair would hurt her feelings or make her feel extremely uncomfortable, you really might want to think twice about saying anything. But if you do say something, be very considerate and say it in a way that won't make her feel awful.
Here is some information about body hair, a body hair story.