Salesforce: Custom Object Search Behavior
Not all fields in Salesforce custom object are searchable in Salesforce. For example, you can refer to this documentation for more information.
If you have a lookup field such as Contact Lookup field, you can implement the following workaround:
1) You have a Contact lookup field in your custom object. This field will only keep track of the Contact ID.
2) Create a new text field (searchable). You don’t have to expose this field in the layout.
3) Create a workflow that copies the Contact Full Name into this Text field.