if it is the socket like that^^^ then the first thing i would try is to reflow the solder at pcb. ive done several where that is all it took as the crappy rohs solder had just cracked. just need to hold a fine tipped soldering iron on each point for a couple seconds.You didn't mention whether you have already tried to use a different monitor, but if you didn't, then you should. If you have already tried more than one monitor and several different cables, then it sounds like the HDMI header is the problem, and that is almost definitely soldered to the motherboard. Open it up to confirm and source the necessary replacement part.
I only have one monitor. I tried with HDTV but it was not detecting connection. Maybe with HDTVs it requires consistent connection without interference. The only way to figure this out is to disassemble laptop, take out motherboard and check HDMI connector.You didn't mention whether you have already tried to use a different monitor, but if you didn't, then you should.