Spoiler

The movie sort of answers that. Another older lady at one point says "Who'd want to hurt poor Martha?" or something like that.

The best way to get the town's attention and let them know he means business is to kill someone defenseless and innocent. If he killed a middle-aged man, people would be slightly less quick to worry. Reasonable he might have been involved in something shady and had it coming. Typically we're less likely to think that way of a little old lady.

Defenseless and innocent really only leaves two options: The elderly and children. And he wanted a child, so no sense in hurting one. So old person it is.

Just to get everyone's attention and get arrested. If you're suggesting she had to have done something to "deserve it" in his eyes, I suppose there could have been something, just wasn't really important for us to figure out.