EMB140 FLT CREW HAS A TXWY EXCURSION WHILE TURNING.

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UPON LNDG AT LAX; WE WERE INSTRUCTED TO TAXI TO OUR GATE VIA TXWY C AND HOLD SHORT OF TXWY B-16. WE MISSED THE HOLD SHORT POS DUE TO RAIN AND FOG CAUSING A LOW VISIBILITY CONDITION. WE IMMEDIATELY NOTIFIED THE ATC CTLR OF THE SITUATION. THE ATC CTLR REQUESTED THAT WE PERFORM A 180 DEG TURN ON THE TXWY AND TAXI VIA A DIFFERENT RTE. AS THE ACFT WAS TURNING; THE NOSEWHEEL SLIPPED AND CAUSED ONE SIDE OF THE ACFT TO SLIP OFF THE TXWY BOUNDARY LINE SLIGHTLY DUE TO THE WET CONDITION OF THE SURFACE. WHEN WE ARRIVED AT THE GATE; WE NOTIFIED MAINT AND ENTERED AN INFO-TO-MAINT ENTRY IN THE MAINT LOG FOR A SAFETY INSPECTION OF THE LNDG GEAR COMPONENTS JUST IN CASE. WE WERE COMPLETING A 13+ HR DUTY DAY; ON OUR 6TH LEG OF THE DAY. WE FLEW IN DODGING WX AND TURB ALL DAY AND HAD TO PERFORM THE ILS TO MINIMUMS AND WERE FATIGUED BY THE LAST LEG. WITH THE LOW VISIBILITY DUE TO WX; WE HAD MISSED THE TXWY SIGN. ALSO; WE HAD UNDERESTIMATED THE CONDITION OF THE TXWY WHILE PERFORMING THE TURN ON THE TXWY. WHEN TAXIING IN LOW-VISIBILITY CONDITIONS; BOTH PLTS SHOULD BE EXTRA VIGILANT WITH THE SURROUNDINGS AT ALL TIMES. THAT MEANS BOTH SETS OF EYES ON THE OUTSIDE. ALSO; IF REQUESTED TO PERFORM A 180 DEG TURN ON THE TXWY OR EVEN RWY WHEN THE SURFACE CONDITIONS ARE WET/SLIPPERY COMBINED WITH LOW VISIBILITY; YOU SHOULD THINK TWICE AND REFUSE IT -- EVEN IF YOU THINK THE TXWY OR RWY IS WIDE ENOUGH. YES; IT WOULD BE EMBARRASSING TO HAVE A TUG COME OUT; BUT THAT IS A LOT BETTER THAN HAVING TO EXPLAIN YOURSELF IF YOU CAUSED ANY TYPE OF DAMAGE TO THE ACFT.