We received and answered this question via email as well, but for anyone else wondering:
this knowledge base article explains how the online status indicator works. The status indicator pings a Windows computer's NetBIOS name from the server hosting the Lansweeper web console. If a Windows computer is marked as offline when it shouldn't be, this is due to the aforementioned ping request failing, likely due to the computer name not resolving to an IP address. This is a DNS issue and not a Lansweeper issue and needs to be resolved on your DNS server.
As for the IP address listed below the computer name not being correct, which is not related to the online status indicator: the IP address will update if you successfully rescan the computer with the Rescan or one of the other scanning options. If you use a scanning method that connects to the computer's name though, this will again require successful DNS resolution of the computer name.