I tried adding a blue flag to one of my planets and it said it was successfully flagged, but it's not really flagged when I close out of the planet info and look at the database. There's no flag next to the planet I supposedly flagged.

Tried removing then re-adding, but no matter what the planet is not flagging (even though the option to remove the flag appears)

It seems like it does not work on this one planet I am trying to flagged, as I was able to flag another planet with no problems.

Maybe this info will help? I momentarily lost it to someone but I took it back and now am guarding it.

My assumption is that this happens when the said planet is alerted. I've had a few occurrences where a flagged planet doesn't show under the "Flagged" filter anymore after I've alerted/conquered it.

I have seen this happen but only to planets I've alerted. Once the alert expires the flag is there (if the last action was to add the flag and not remove the flag).

Ugh. I get this too! It doesn't really bother me too much unless I flag an sNAP'd planet and it doesn't show up and then bad stuffs happen.
But on a serious note: you're not the only one who gets this.