How should I respond to this

I would probably ask if she's okay first... Maybe she just wants your attention. People can really be strange and there's no way to know why she's doing that.

I would just ignore it as some people are just odd.

Coughing out loud can be any number of reasons when you walk by. It could be the way you are dressed or the fact that you ignoring her and she might fancy you. There is nothing wrong with women of any age trying to express themselves to someone and get their attention. Clearly, her coughing is getting your attention in some way. If you don't like it be an adult go and talk to her and ask her yourself. Ignoring things doesn't change things maybe all she wants to do is be your friend it's okay for neighbors to be friends. You might have a feeling of being unapproachable. My husband tells me from afar I'm intimidating so it makes it hard for some people to come up to me till we have talked to each other.

You could respond by saying hello and asking her if she is okay. Or you could studiously ignore her, not even look her way, and see if the behavior stops. If I were in your situation, I would probably say hello and ask if she was okay.

It sounds like she wants attention but it is odd behaviour

Maybe it's a code for you to greet her first. You know women have complicated codes.