I want to actually put forth one-two month for the process of candidate picking, electing, presentation, and "inaugruation" (but will go for the lowest month level that has a fair shot of actually winning if need be).
Why one-two month(s)? Well, real life happens, and some months are busier than others. I'm talking August-September (beginning of school for some) and December-January (Holidays and School). There shouldn't be any real-life-styled campaigning for this sort of thing. We can get to rules and regulations on how candidates can present themselves, but we don't want to turn these elections into a political match lit whenever.
If a Council member is doing good, then they'll get re-elected; plain and simple. Stability would only be a major issue if the Council members are seen not to be up to snuff by the end of the month, in which case we wouldn't want them to be staying in for months at a time, especially if there is not an impeachment process in place (or rather, if we choose to even have one).
Being on the Council should be a prize of sorts that doesn't have the chance of creating an oligarchic of sorts. If frequent rotation is needed that should be given as an option; if not, incumbents can get re-elected no sweat.
Truthfully though, the actual process of elections (how long are they going to be permitted to be "campaigning", what kind of "campaigning" is even permitted, etc.) is a really important factor in all of this.
Now at the end, I think I want to put a firm vote on 2 months (cause I don't see too many people agreeing to 1 month, plus the full election process could take quite alot of time if lengthy stuff is allowed).