There are two types of petcock valve, a cone type and a flat plate type. Easy to tell apart, the cone shaped housing gives that type away.
Probably you have the flat plate model. There's a cover plate with two screws or "rivet" things holding it down. Drill off the "rivet" heads if it has those, unscrew the screws if it has screws. The lever part and a serpentine spring will come out, underneath is a rubber disk part with 4 holes: one is on a keying pin, one goes to the hose nipples, one goes to the "on" inlet tube, last goes to the reserve inlet.
To renovate, check the rubber disk for wear. There are (were?) slightly raised rings around the holes, if worn to flat the disk is not too good anymore. Some are reversible so you can flip it and get a virgin surface. They are available on ebay and such as replacement parts but I don't think Honda ever sold them separately.
The lever part is pretty much flat except for fuel channels, these line up with the rubber disk holes to let fuel flow from tank to nipples.
Polishing (lapping, really) the surface flat and smooth helps a lot. Just some metal polish on a flat plate (Solvol is excellent) will shine and flatten the surface. I have seen badly corroded ones on bikes left years outside, this is probably unrepairable.
With needlenose pliers, make the spring bends higher for more tension.
Lube the friction areas with whatever lube you like, I use Dow 111 silicon o-ring lube for that kind of thing.
If you drilled off the "rivet" heads: these are obviously not rivets, just peened over teats in the housing. Get a 4-40 tap drill and tap and two round head 4-40 screws, - stainless is good, brass is not good as electrochemical corrosion will attack them. Drill straight through the case using the first drill mark as a centre to align the drill. Tap all the way through: use some lube (tapping fluid, WD40, 3 in 1, whatever you have). The metal is super easy to tap but be careful not to snap the tap off: 4-40 taps are quite delicate.
Clean the case and blow out the holes. Reassemble with lube. Use a bit of antiseize on even SS screws.
It should turn much smoother and not leak.
The cone type is totally different except for the cover screws. Not much can be done to revive a really worn out one, a bit more tension on the spring plus careful cleaning - particularly the bore and plastic valving cone - and adding some lube may stop any leakage. It has for me anyway. Reassembly is trickier - the valving cone has to be keyed correctly to the lever - but it isn't rocket surgery.