WHILE EN ROUTE, THE COMPANY DISPATCHER NOTIFIED AMERICAN FLT 844 (VIA ACARS) THAT HARLINGEN (HRL) HAD UNFORECAST FOG & WAS 300' OBSCD WITH 1/2 MI VIS. AT 2335, AS THE FLT BGN A DSCNT FM FL 310, THE SAME CEILING & VIS WERE RPRTD. HOWEVER, AS THE FLT WAS CLRD FOR AN ILS RWY 17R APCH, HOUSTON ARTCC ADZD THE WX WAS (IN PART) 200' OBSCD, 3/8 MI VIS WITH FOG. WXMINS FOR THE APCH WERE 200' CEILING & 1/2 MI VIS. TWO APCHS WERE MADE USING THE AUTOPLT, BUT BOTH WERE TERMINATED WITH MISSED APCHS DUE TO WX. THE CREW STARTED TO DIVERT, BUT RECD A WX UPDATE (VIA ACARS) THAT THE WX WAS IMPROVING, SO A 3RDILS WAS ATMTD. AFTER ACTIVATING RWY LGTS, THE CAPT HAND FLEW THE APCH. AS HE TRANSITIONED TO A VISUAL APCH, THE DSCNT RATE INCREASED & THE ACFT TOUCHED DWN SHORT OF THE RWY, DMGG THE LNDG GEAR & APCH LGTS. THE CREW MADE A GO-ARND, THEN DIVERTED WITH NO FURTHER INCIDENT. A COMPANY WX OBSERVER AT HRL NOTED THE 0010 WX WAS 200' OBSCD & 1/2 MI VIS & THE 0018WX WAS 100' OBSCD & 1/4 MI VIS, BUT THE CREW DID NOT CONTACT THE OBSERVER, AS REQD, BFR MAKING THE APCHS.