Well, I think that a lot of younger Japanese people tend to be shy and they sort of make it as quick as possible. So yes, just your name and a smile would be fine, and the other party would think, "Well, that was a polite girl to have greeted us so patiently."
But if you are at a more responsible age, as suggested, a typical greeting should include asking whether there are rules you need to keep in mind such as things about garbage. If you have already lived there for some time, you can ask the typical, "gomeiwaku o okakeshite imasenka?" as a courtesy pose to make sure you're not annoying anyone. Also typically, people might find this as a good chance to ask about the neighborhood in a "oh, by the way, I have been wondering" tone. About shopping or doctors etc.
Actually, the idea is supposed to be that you want to make it quick, because you know that the person needs to go to greet other neighbors and you don't want to let the person just standing out there in the cold whether. But it's not that there is no friendship to follow. Typically, if you meet each other in the elevator you might talk about the weather, and if one of you are with a small child you might try to be nice and chat with the kid, and if you feel you have something mutual you might try to talk about that. The idea is that if you have done the greeting, it's easier to say, "Oh hi, you are that person from the other day."
However, you have to admit that if the other party is young and single, or even foreign for that matter, it's hard to find a theme you can chat about. You don't know their hobbies, you don't know their occupation, you don't know what kind of family they have back home, and mostly locals tend to think that young people don't like to talk. So it would be encouraging if the younger person or the foreign person starts the small talk. For example, a lot of people tend to ask where they lived before moving in. That would be an easy conversation.
It's true that conversations tend to go round and round once you start doing that rotation. The trick is to not repeat what you've already said, and the other trick is to just shut up. People want to look polite, so they don't want to be the first one to end the conversation:)
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