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';
<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.