If you are looking for deep things to say to a girl as a way of introducing yourself and allowing her to get to know you, you are playing a dangerous game. While finding deep things to say to a girl might seem like a great way to impress her and attract her to you, it tends to make it more difficult for you to achieve what you want.
Saying these kinds of things to women off the bat tends to work against you by intimidating her and making her afraid to talk to you (so as not to appear dumb or boring), by making you seem vein or self-conscious – a truly confident man wouldn’t need to validate himself by saying these kinds of things right off the bat – and also because they tend to be impersonal and abstract, things that could be said to any girl and things that would tend not to make her feel special.
Generally, finding deep things to say to a girl should come later, after you have gotten to know her fairly well, and after she is comfortable around you and feels like you already respect and appreciate her as a person.
There is, however, a way to talk about something beyond the general shallow conversation starters that are supposed to work on girls. Sure, complimenting her appearance, making a simple comment about something she said, or simply introducing yourself confidently is always an effective strategy, but there are ways to achieve similar ends while skipping a lot of shallow and undesirable conversation. This is done, above all, by showing a strong amount of passion about the things you decide to bring up to her.
To be clear, the non-shallow things you talk to her about still should not be abstract, theoretical concepts that you find interesting or consider yourself knowledgeable about. They should be things that are not removed from the direct world around you. They should be a certain cause that you are invested in, a certain job or activity that really inspires or interests you, etc. Women are attracted to men who they see as passionate about something, as they associate that with a higher likelihood that you’ll make a good lover, a good father, and a good boyfriend/husband.
You can very much get away with talking passionately about, say, the foundation you volunteer for, or the music that you play, etc. The key here is to come off as completely and totally genuine. Never make it seem like you are pretending you care so that she will think you do, or so you have something to talk to her about. Say things like, “Sorry, I may have gotten carried away, but I’m sure you don’t care about that so much…” and then move to introducing yourself and talking about other things.
In this way you can talk to girls about things beyond the petty, shallow conversation starters that are often used, while not using some of the deep things to say to girls that might turn her off from you. Hopefully, you can in this way get the best of both worlds – a conversation with a girl that both works to attract her and make her want to continue talking to you, and a conversation that is interesting.