B717-200 CREW ENTERED THE WRONG TXWY AND WENT TO THE WRONG RWY BECAUSE OF REDUCED VISIBILITY RESTRS AND EXTREME CREW FATIGUE CAUSED BY REPEATED CREW 'STAND UP OVERNIGHTS.'

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PUSHED BACK FROM GATE; GIVEN 'TAXI TO RWY 7 VIA TXWYS J AND A.' AS I STARTED THE TAXI; I COULDN'T MAKE OUT TXWY J. VISIBILITY WAS ADVERTISED 6 SM ON ATIS; I ASKED GND AND THEY REPLIED CURRENT VISIBILITY 1/4 SM. TURNED ON A TXWY I THOUGHT WAS TXWY J VIA THE ARPT DIAGRAM (IT WASN'T LABELED) AND IT TURNED OUT TO BE TXWY H. I STOPPED AND WE ASKED GND IF WE COULD USE TXWY H (APPROVED). AT THE INTXN OF TXWYS H AND A; I TURNED R INSTEAD OF L. VISIBILITY WAS BELOW MINIMUMS SO WE NEEDED A TKOF ALTERNATE. WE CALLED DISPATCH AND RECEIVED MCO. UPON REACHING RWY 25; SWITCHED TO TWR; AND I DID A RWY CHK. NOTICED DIRECTIONAL GYROSCOPE AND RWY SIGN AND REALIZED WE TAXIED TO THE WRONG RWY. NOTIFIED ATC AND WERE APPROVED FOR TKOF ON RWY 25. I BELIEVE 3 FACTORS (AT LEAST) WERE CONTRIBUTING: 1) THIS WAS A CONTINUOUS DUTY OVERNIGHT -- ARRIVED AT XA45 PREVIOUS NIGHT AND HAD AN XH20 DEP (4 1/2 HRS SLEEP!) 'FATIGUE.' 2) 'VISIBILITY' MADE TAXI DIFFICULT TO SEE SIGNAGE -- 1/4 SM (GND CTLR COULD NOT SEE US TAXI WRONG WAY). 3) 'PREOCCUPATION' WITH THE WX AND GETTING A TKOF ALTERNATE. THIS WAS THE 3RD CONTINUOUS DUTY OVERNIGHT IN A ROW; I WAS VERY FATIGUED AND IT AFFECTED MY AWARENESS; JUDGEMENT AND I DIDN'T NOTICE ERRORS I WAS MAKING.