What should I enter for locality name if on my w2 it just says "local tax" in that box?

I'd recommend checking with your employer's payroll department to see what "local tax" they collected (withheld) from you  - they should be able to give you the name of the locality.

If the locality isn't listed on your W-2, the TurboTax program may not give you proper credit for the payment or it may fail to prepare the appropriate city tax entry for your state/local returns.

Some local tax authorities do not require you to file a return and may even charge a fixed or flat tax. In those cases, it may not make any difference if the locality is entered in the W-2 interview.