I'm not sure whether it's possible to get the monitor name in a standard SNMP alert. You might be able to create a custom SNMP trap based on the log message which contains the monitor name. Try searching AskF5.com and/or this forum for "custom snmp trap" for some examples.
SOL10966 notes a request for enhancement CR, CR82787, to support the logging of the monitor name in the logs. You might open a case with F5 Support referencing this CR and ask if it already includes SNMP traps in the RFE. If not, they could add this feature to the request.
SOL10966: Determining which monitor triggered a change in the availability of a node or pool member
https://support.f5.com/kb/en-us/solutions/public/10000/900/sol10966.html
Aaron
Nimbostratus
