In the early 1960s, a young mother’s anxiety makes her suspicious of everyone, even her 11 year-old son, who takes on a mission to heal her by making her laugh.

I love the idea. Can you rewrite the logline by focusing more on the main character (the son)? What specific event made him decide to take action? And given that healing is a process, what is the specific point the audience will be looking forward to, i.e. what will indicate that he is successful? Most great story concepts allow us to visualise what a positive outcome would look like.

Am I assuming right that this is to be a comedy?

Where will the comedy come from? Because anxiety is not funny... and if we're going to make the mother laugh, the situations better be funny to the audience, too.

Those are few off-the-cough thoughts. I hope this is helpful!