How do I adjust an incorrect cost basis for a stock?

a)  Make sure you understand what "cost basis" is.  See the Quicken Help glossary.  The same definition applies whether the account holding the security is taxable or tax deferred (IRA, et al.).  Some financial institutions vary cost basis depending on account type.  Cost basis is also computed on a security level -- the cost basis for an "account" is the total of the cost basis for all securities held in the account.

b)  If you are clear on the definition and specific numbers, the cleanest way to adjust the cost basis is with 1) a Remove Shares taking out the shares with the erroneous basis, and 2) an Add Shares that adds those shares back in with their correct cost basis.  The Add Shares should also reference the correct acquisition date for the shares being brought back in.