This is not a solution. More a steps to help you diagnose (at the time of writing, not enough information to offer a solution).
So these are the steps I would follow to see what may be the cause.
First start the Notes client in Basic mode. You can do this by adding the
-sa to the command line of the shortcut. Straight away this will tell you if the issue is in the core or eclipse part of the application.
If the issue no longer occurs then the issue is in the Eclipse part.
Issue in core area.
Determine the steps leading up to reproducing the issue (eg. Runs idle, or user clicks something).
Check the Notes.ini for the setting
EXTMGR_ADDINS. If this is set to anything, blank it out and test. This will rule out 3rd party add-ins.
While the handles are opening go to the START menu and select:
Lotus Applications->Support->Create NSD and continue (If I recall correctly). If that option doesn't exist, just run NSD from the Notes folder. Do this 3 times (take a little time between each one).
Open the NSD file in the IBM_TECHNICAL_SUPPORT folder and look for the
Handles by Count. This will give you the top 10. Compare these against the other NSDs to see if handles are closing or remaining opening. You will need to post the process information above to give more details.
Issue in Eclipse area.
Start by removing the Workspace folder (with client stopped. Back it up). This will rule out any third party plugins or settings the user may have set.
If the issue persists I would recommend to look at the workspace\logs files to see if you are getting any WARNING or SEVERE errors in the logs. There will normally be stacks associated with them. If you post the stack I can suggest further.
Other things to try.
Disable policies for the user and try on a clean client. If the issue continues, then check what is changing in the policy settings. You can set each one by one for this.
In the Notes.ini for the client add CLIENT_CLOCK=3 . This will give more information in the IBM_TECHNICAL_SUPPORT folder console logs.
I'd recommend testing on the latest fix pack as well to rule out any existing issues.