Hey rb51, some info that may help:
First, there is shell access to the LEM appliance. You can use an SSH client (like PuTTy) to access the LEM. Details on how to do this are here:
If you connect to this shell, under the MANAGER options you can run a VIEWSYSINFO. What version is the LEM? How much memory does it have? What platform is it on (HyperV, VMWare, physical)? What are you memory reservations set to? Memory is really important to the LEM (see this other thing I wrote: Best Practices for LEM Memory Allocation) so that's where I usually look when the LEM seems to have regular crashes. If configured correctly, the LEM should be able to run indefinitely without needing a reboot.
Also, do you have the Reports console installed? If so, can you run a Database Maintenance Report? How many events are you getting per day, on average? What are the most common events? Is something flooding the LEM or was there a spike in traffic recently?