At the moment it is not possible to rename an option of a custom field, e.g. a Select List custom field. This is fairly easy to do using SQL. Please shutdown JIRA then execute:

update customfieldoption set customvalue = 'New Option' where CUSTOMFIELD = <cfid> and id = <id>;

To rename the option. Where <cfid> is the id of the custom field and <id> is the id of the option you would like to rename.

To get a list of all custom fields do:

select * from customfield;

Then update all issues with this value:

update customfieldvalue set STRINGVALUE = 'New Option' where CUSTOMFIELD = <cfid> and STRINGVALUE = 'Old Value';

Replace <cfid> with the custom field's id and 'Old Value' with the text value of the option.

Then restart JIRA and re-index the data (Administration -> System -> Indexing).

For details on editing the custom field tables, see the custom field tables documentation.