A B737-700 APU OIL PRESSURE INDICATED LOW IN FLT DURING SINGLE GEN OPERATION. THE CREW SECURED THE APU; DECLARED AN EMERGENCY; AND DIVERTED A NEARBY ARPT.

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ACFT HAD A MEL FOR #1 ENG GENERATOR INOP. OVER ZZZ; WE NOTICED THE APU LOW OIL PRESSURE LIGHT STARTED TO FLICKER. IT WOULD NOT REMAIN ON BUT JUST BLINK ON INTERMITTENTLY. WE CONSULTED THE QRH WHICH SAID AN APU LOW OIL PRESSURE CONDITION WOULD CAUSE THE APU TO AUTO SHUTDOWN. THE APU WAS RUNNING WITH NO OTHER LIGHTS (MAINT) ON AND NO TEMP FLUCTUATIONS. WE BELIEVED IT MIGHT BE A FALSE INDICATION AS THE LIGHT WOULD JUST FLICKER AND NO AUTO SHUTDOWN WAS OCCURRING. I MADE VOICE CONTACT WITH DISPATCH AND EXPLAINED WHAT WAS GOING ON AND TO RELAY TO MAINT. THE FO CHKED ALL WX DOWN-RANGE IN CASE THE APU DID SHUT DOWN AND IT WAS CLR THE REST OF THE WAY. WE RECEIVED SEVERAL ACARS MESSAGES SHORTLY THEREAFTER CONCERNING NOTES IN THE QRH AND IF THE APU WAS SHUT DOWN. I MADE VOICE CONTACT AGAIN AND ASKED DISPATCH TO GET A CHIEF PLT ON LINE TO GET ANOTHER OPINION. THERE WERE NUMEROUS PROBS GETTING THAT TO HAPPEN AND WE EVENTUALLY LOST VOICE CONTACT WITHOUT TALKING TO A CHIEF PLT. I CONTINUED VOICE CONTACT ATTEMPTS. WE RECEIVED AN ACARS MESSAGE RECOMMENDING WE DIVERT TO NEAREST SUITABLE ARPT. WE PICKED ZZZ1; TOLD ATC AND SENT DISPATCH OUR INTENTIONS. DISPATCH SENT BACK A MESSAGE THAT ZZZ2 WAS NEAREST SUITABLE ARPT; SO WE HEADED THERE. DURING THIS TIME; IT WAS OUR IMPRESSION THAT IF A DIVERT WAS RECOMMENDED; THEN THE CHIEF PLT MUST HAVE BEEN OF THE OPINION THAT THE APU SHOULD BE SHUT DOWN (ALTHOUGH SOME OF OUR CONVERSATIONS WITH DISPATCH LED US TO BELIEVE THAT THEY THOUGHT THE APU HAD ALREADY SHUT DOWN). WE SHUT DOWN THE APU AND ACCOMPLISHED THE APU LOW OIL PRESSURE AND SOURCE OFF CHKLISTS IN THE QRH. WE LANDED IN ZZZ2; 500 LBS OVER STRUCTURAL LIMIT; ABOUT 10 MINS AFTER THE DIVERT MESSAGE WAS RECEIVED. THE INABILITY TO TALK TO A CHIEF PLT AND MAINTAIN VOICE COM WITH DISPATCH DEFINITELY AFFECTED OUR DECISION MAKING PROCESS.