Answer to this problem is spread across the internet in bits and pieces. I am just putting it in here under the context of my problem.
We have a lot of migrated sites from SharePoint 2010 to SharePoint 2013. For custom lists “Find an Item” search box seemed to work just fine. However, there are calendar lists. In such lists, there is a Location field which is of type Single line of text. We could not seem to search entries in this field reliably. However, if I create a new column called Location1, I can search that field just fine. So what is going on? Why does search work for some columns and not work for some other predefined columns? Most importantly what can be done to resolve it. I had created a post for the SharePoint 2013 admin forum. I tested things that were recommended. But no luck. The post is here:
The resolution was to make the related crawled property searchable in the search service. For this, click on your search service application on the central admin>
Select Search Schema in the left menu as shown below: Edit if needed for it to be searchable.