I am using a calculated field in a form (Data Type = Single Line of Text), however I would like to make this field editable for a specific users having "System Administrator" roles.
I tried using the field security profile to make this field editable (so that System Admin can update or create records here) however update/create options are disabled here in security profile and cannot be altered.
Is there an appropriate way to achieve this? Can we make a calculated field editable for specific set of roles?

No, you cannot directly edit the value of a calculated field.
A work-around would be to create a separate field, where only specific users can enter data. In your calculated field you could then add a condition to take either the manually entered value from the other field (if it exists) and otherwise use the original action for the calculated field.