In Spain and other spanish speaking countries in latinamerica (most of them): "Déjale un mensaje en la contestadora"
In Argentina (we speak a little different):
"Dejale un mensaje en el contestador"
Note that in spanish we don't specify the gender of the person who is going to receive the message ("him"). And in English you say "leave a message on the phone" but in spanish we use to "leave a message in the answering machine", "answering machine"="contestador" or "contestadora". The difference in the two options is the gender of the "machine", in some countries we say "contestador" (male subject) and in other countries "contestadora" is better (female subject).
The Spain (and other countries) Spanish version could have some variations, I'm from Argentina.
Telefono
There are several ways to tell someone to leave in Spanish. One can say dejar. Another way is to say salir. Yet a third way is to say abandonar.
licencia entoces
Voy a irme
You say bye or i have to leave
celular or móvil
condenado telefono
Tele'fono mo'vil
my phone has died = mi teléfono ha muerto
nunca me dejes
Alístate para ir.
No salgas -or- no te vayas