Hi Sai,
If you change the name of a custom field which are used in several projects, this will affect all projects.
There is no way of having the name change only affect one of the projects where it's used.
If you want to retain the current values in issues where this field is used, Then what you can do, is to create a new custom field of the same type, with the same values (if needed/applicable) and with the name you want, and apply this to the project.
After this, you can do a CSV export of the issues from the project where the new custom field is applied, and the values they have in the field that is to be renamed.
Thereafter, you can import the CSV file again, mapping the values to the field with the new name.
This is described in much closer detail, in the accepted answer here:
https://community.atlassian.com/t5/Jira-questions/A-way-to-bulk-copy-custom-field-values-from-one-custom-field-to/qaq-p/323932
I would suggest also reading up on the documentation about CSV export/import, and doing a test before proceeding with this.
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