String relief groove?? For pinned bridges?

I have a problem with the third string on a pinned bridge ukulele. It makes a clicking noise when picked with my thumb. I loosened the string and easily removed the pin, the groove in the pin was aligned with the string. In the past on a different uke the pins flew out when installed this way, however on the current uke the pin will not seat when the groove in the pin does not align with the string. So should there be a groove for the string in the bridge pin hole?? Or should I be looking somewhere else?

There are two ways to handle pinned bridges.

The most common is with bridge pins that are slotted. That slot lines up with the string and is intended to be deep enough that the string nests in it. If i't not then the pin won't seat all the way because of the interference.

The other way is to have un-slotted pins (or turn the pin so that the slot is not lined up with the string) and to then make the relief in the bridge itself.

The slotted pins came about because it was a quick and dirty way to accomplish the job at hand, and didn't require a skilled person to fit the strings to the bridge. However it's a really poor way to secure the strings, and while unlikely to cause much damage on the lower tension of a uke, on steel string guitars it's the leading cause of damage to the bridge plate.

With un-slotted pins and the string being fit into the bridge, the string will be firmly anchored when under tension, and you should be able to pull all the pins out without the strings coming loose. The knot or bead on the end of the string is pulling up against the bridge patch and the pin really isn't doing anything other than filling the hole.

Try picking the strings instead of the bridge pin (kidding)

Seriously though.....I'm not sure what you mean. The pin makes a clicking noise when you pluck the 3rd string?? Or the pin is loose in the hole and can move around making noise??
Bridge pins are tapered and the pin hole should be tapered to match giving a snug fit when you push it in. This is the same for slotted or unslotted pins. Sometimes the winter dryness can make the pins loose or sometimes the factory can over-ream the hole and the pin won't seat well. If the pin used to work well I'd suspect dryness and get a humidifer

There was(is) an extraneous noise when I picked the third string with my thumb in the bridge area (no noise with index finger). I had the ukulele tuned to Bb at the time, I've since gone back to C tuning and it seems to have abated for the most part. Not exactly sure but think it may have been more from movement of the string in the pin slot than movement of the pin itself, though the pin was not tight when I went to remove with the string slack -have not attempted to remove with string under tension. Only have past experience with one other pinned bridge (it had string relief grooves in pin holes) and they were very snug even with no string tension.

Are you sure that this "extraneous noise" or "clicking" is not due to the third or C string (thickest) beating against your frets? Maybe your nut/bridge is a little low? I've found that even on a medium action, if you really wang the string (or dig in with your thumb) the rotational vibration of the third string will sometimes bang against the frets creating an "extraneous noise". It is sort of part and parcel to playing a short scale length instrument like an uke. That you had the string tuned to a lower tension in a Bb tuning (and thus a greater arc on on string vibration) would make this problem more obvious. Maybe it isn't the pin at all but too low an action for your playing style? Just a thought.