Sorry, this is long, but I hope you wiser and more experienced parents will be willing to help.
Our next door neighbors moved in two years ago, and we've all been friendly. Last summer, the adult daughter moved in with her little girl who is a few months younger than my older son. DS was very excited to have a child next door and hoped to play together. We would frequently see the little girl watching us from a window when we were outside playing. I took DS and knocked on the door a few times to ask if the girl could come out to play, but it never worked out. Once he played with her for a few minutes when she was out front with her uncle. Another time we saw her watching us from the front door, and I helped DS cut a handful of daisies and take them over. The girl seemed to want to play, and her mother acted cordial, but nothing. It appeared that the parents were separated and made up, and the little girl and mother moved back out. I figured the mother had enough on her plate through that time, and probably wasn't up to trying to deal with playdates with new friends. No problem. But they moved back this winter and seem to be here to stay. Now that the weather is nice, we are starting to see them outside more, and DS would love to play. But I think the mother tries to avoid us. She barely says hello when we pass coming or going. And last night as we arrived home, they were out front with sidewalk chalk. While we parked, she picked up the little girl and went inside. Could have been coincidental but didn't feel like it.
Should I just give up? Should I try again? It seems like such a shame to have a child next door and not play. I didn't grow up with neighborhood friends and am at a loss for how to handle this kind of thing.