Open Keychain Access in Applications/Utilities
Search for your Skype account.
Delete all entries. The Keychain isn't smart and will keep all old entries when you add/change a password.
While you are in the Keychain run Keychain First Aid,
Choose Keychain Access > Keychain First Aid.
• Enter your user name and password.
• Select Repair and click Start.
• If any problems are found run again to verify all repaired.
Open Skype and enter the password in your account.
This should stick now.
It's possible the Skype preference file in your User's Library**/Preferences folder is corrupt.
Delete the com.skype.skype.plist
Log out under the Apple in the Menu bar.
Open Skype and add password. It will be saved in a new plist file.
**Show User's Library
The User's Library folder is hidden by default. To unhide: Select the Finder in the Dock. Under Go in the Menu bar > hold down the Option key and you’ll see the Library.
Finding the User Library Folder
http://www.takecontrolbooks.com/resources/0167/site/chap11.html#FindingtheUserLi braryFolder