In LCRM, field-level security must be set to Visible and not Read Only for at least the Luminate CRM API Only profile.

The LCRM field can only be of certain types, and you must be attempting to match it to a field of that type. For example, a checkbox field in LCRM can only be matched to a boolean field in LO.  See What Luminate CRM data types can I map to fields in Luminate Online?

If the field is part of a managed package, the Integration User may need to have a license to that package.

In LO, the field must be set to Admin Can Change. See Custom Fields cannot be mapped in Database Configuration.