Follow this guide to make a custom cotton swab.

You can also use dental picks with barbed tips (non-flavored).

You can use a normal cotton swab, but most are too thick and have loose cotton that can snag on sharp edges.

Apply a few drops of highly-concentrated isopropyl alcohol (>90%) to the cotton.

Insert your tool between the middle contact pad and the perimeter of the HDMI port itself.

Don't force the tool into the port, as you risk breaking the middle contact pad.

Slide the tool along the perimeter of the port to clean it.

Wait a few minutes for the isopropyl alcohol to evaporate before plugging in your device and testing it.