<p>in Web 2.0 applications many users usually want to stay logged in ('remember me' flag) and on the other hand their cookie can give access to very private data. Is there a way to prevent that somebody who steals the cookie - directly from the computer or via sniffing - can use the cookie to get access to the user's data? Always HTTPS is not an option.</p>

<p>Thanks, Bernd</p>

<p>[Edit] Connect the IP address to the cookie is not an option either.</p>

Store a cookie that is an obscure ID into your local server database. Do a server-side DB lookup based on the ID provided in the cookie.  Be sure to make the ID sufficiently complex that it cannot be easily guessed.  Map the ID to the IP Address of the user. If their IP changes, then force them to log in again, and create a new ID.
On second read, it sounds like you want a high level of security with your hands tied. The user must have the choice to remain logged in and thus increase his/her risk.  You can implement all the security in the world from the application's and server's point of view, but if the user forgets their laptop on a table in Tim Horton's (Canadian Starbucks), then none of it will do you any good.
Leave the choice up to the user as to whether or not they remain logged in, and give them a warning about their information being at risk.