Windows 7 "Could not reconnect all network drives" issue

If the 2k3 server is a domain controller or a member server in a domain make sure that the Win7 HomePremium machine has a workgroup name that reflects this. If the domain is named ThisIsMyDomain.lan then make sure the Win7 HOmePremium machine's workgroup name is ThisIsMyDomain and exclude the .lan suffix. Other wise if it's just a workgroup server just make sure the workgroups names are the same.
Then make sure that what ever UserName and PassWord you are using to log on to the Win7 HomePremium machine is also present as a username and password on the 2k3 server, either within AD users and computers or as a local account on the server, making sure that that user has appropriate permissions to the shares.

Both, laptop and server are in same workgroup. There is no domain,nor AD. I dont use the
same username and password to log on to windows and to access network share on windows 2003 server.
Actualy I log on to windows on laptop without password.
To access network share,I am using username and password which I created on server. It works
well when I manually type username and password but as soon as I reboot my laptop connection
gets lost and require to type my credentials again to access network share.
What is the order of credentials that Windows uses when access to network share
after reboot? Does it first try to use local account on my laptop which I used to log on to Win
or another one that I saved when accessed network share?

Yes, it will probably attempt first to default to what we use to call pass thru authentication using the account and password you are coming from, if that is present on the resource machine things should work automagically.
Your problem may very well be with a lack of any password, just a guess but perhaps because of the missing password Win7 is unsuccessful adding the account to the local credential vault and as a result when the resource machine queries to confirm you access token, it's not working properly, and you have to re-enter it manually.
Try logging on to the Win7 machine with a known good account on the 2k3 server and see if everything works ok, and or add a password for your login and create the account on the server with the appropriate permissions and see if that works ok. At least that may give you a hint as to what's happening. But I know there are and have been known issues with Windows networking using blank passwords ever since XP Sp3, so let us know what you decide as to how to move forward.

Thank you for posting back and updating your thread. Welcome to the forums and we hope to continue to see you around.