In this topic, I'm going to write out what appears to me to be the ineffectiveness of the Godfather's various protections, and what I would do to make them a little more effective. If you don't care about my reasoning, skip to the TL;DR at the bottom.
The first protective option is "cannot be killed at night". This option would be a huge advantage for any non-mafia class, but as you all know, the mafia are usually the ones doing the killing. It protects the Godfather from Vigilantes, Serial Killers, and Arsonists, which is terrific in games with these players. However, if it is a Vigilante, he immediately knows that you are a Godfather, Survivor, Citizen, Serial Killer, or Arsonist, or being protected by a Doctor, and in many games where there are no Citizens and neutrals are lynched just to be safe, this means death anyway. Communication with a Doctor or Sheriff/Investigator, if necessary, will just delay your death. If there is an Arsonist, there's a good chance that his kill will go through night immunity. I think this is default. And Jailors will still kill the Godfather if they have a good hunch. D'oh.
So to me, this option mostly just protects the Godfather from being randomly killed by a Serial Killer in games that have them, which is nice, but let's continue.
The second protective option is "immune to detection". This REALLY means, "you appear to be a Citizen, Jester, Godfather, or Framer" to an Investigator, and "you could possibly be innocent or the Godfather" to a Sheriff. Let's assume that we aren't playing with Citizens, and we are playing with Investigators, as with most games I've played. This means that the best means for a Godfather to protect himself in the case of being investigated (which will happen if the Sheriff says his findings) is to act as a Jester the whole game. If he suddenly claims Jester during his trial, people will obviously lynch. Acting like a Jester may very easily get you lynched anyway, as opposed to more subtle strategies you could otherwise employ. So this isn't a very good strategy at all, and most likely, if you get targeted, you're getting lynched.
For completeness, let's assume "immune to detection" is off. A Sheriff will know you instantly, an Investigator will know you as a Mason Leader (who people rarely put in games), a Mayor (who can reveal himself), or the Godfather (yep, it's this one). You have nothing to claim here. The benefit of this mode is that if the Investigator targets an actual Mayor first, he will be forced to reveal, and either him or the Investigator can be killed by you the next night (though pick well, because one or both of them will be Doctored). It seems to me that being immune to detection is more beneficial currently, because the Sheriff protection is the most useful of anything mentioned so far.
TL;DR: Now, what I would do to amend this big wall of text. I would make a Godfather under the "immune to detection" option appear to be a specific role to Investigators which could possibly be town. I think the Investigator descriptions need a bit of working around to begin with, but simply making it "spends the night with people" could allow the Godfather to claim Escort or Bodyguard.
With this change, the Godfather is much safer from being detected. Against a Tracker/Watcher or whatever they're called now, he can claim that he visited a townie with a benign role. Escort if they died, Bodyguard if they didn't. Smart people will eventually be able to piece together clues which lead to the Godfather's possible lynching, but at least "immune to detection" means something close to what it says.
Now, for "cannot be killed at night". If a Vigilante targets you with the above change, you can now claim to be healed by a Doctor, and investigations may follow through. This now means something close to what it says as well.
I am not claiming that the Godfather is overpowered or underpowered, especially since these options are optional! I just think that the host should have the power to use these options as they are described if he wants to. A small change for such a big wall of text, but let me know what you think!
Side suggestion: I currently can't make a game that has Random Mafia and a Mafioso and can't possibly have a Godfather, without also removing other power roles. If this is possible let me know. However a lot of setups are made impossible because you can't just restrict specific roles through a separate menu. This should be possible. I would use more Basic Towns if i could restrict them from being Doctors, and nothing else. The Basic/Advanced thing right now is somewhat arbitrary and unhelpful to hosts.
Side suggestion: List the roles for "your target is not suspicious" when this message is received by an Investigator. It really is pointing to certain roles just like every other description.