I'm not running for office. I don't see anything that stands out to me as a clue in the day 1 post that hasn't already been brought up. Since the focus of day 1 is on the elected roles though, I think it'd be worth analyzing exactly what the benefits of the roles are.
First the facts, which I am re-typing for convenience in reading my post.
Both roles have bodyguard protection. Both roles are immune to rolechecks.
Mayor gets the day 1 lynch and extra voting.
Sheriff gets 2 incarcerations.
Both roles get an intangible trustworthiness benefit, probably.
Okay. Now let's examine what this means for the town: how would it benefit the town to have a non-red elected, and how would it hurt the town to have a red elected? Since both roles are immune to rolechecks, the only way we can determine the alignment of our elected roles is through behavior analysis or clue analysis (assuming the elected is red, of course)
Bodyguard protection is a huge benefit if we get strong, town-aligned players elected--obviously. Keeps one/two great targets safe from the mafia. In addition, the bodyguard protection is not very beneficial to the mafia (it only protects from Vigi hits, and hopefully the whole town would see the Mayor/Sheriff bleeding red).
The immunity to rolechecks is a minus to the town period, since only the town benefits from rolechecks to begin with. Probably a necessary counterbalance to the bodyguard protection though :p
The mayorial day 1 lynch is largely a wash for the town. Only weak clues, no DT information, not as much time for behavior analysis unless someone blows up (see nemY in Mafia VIII). Not likely to be better than having the town vote on a candidate to lynch. It's a slight benefit to the mafia if they get elected Mayor since they have no risk of a day 1 Mafia lynching. It does potentially give the town information on the Mayor's alignment though.
The extra votes from the Mayor seem less important early on and a potentially huge benefit in the endgame if the game is otherwise fairly even. Mafia can win with a couple extra townies alive if a Mafia mayor is still alive--look at last game and mentally put Qatol back in as mayor and take a look at how hopeless townies ever getting a mafia lynched was at the end if you need evidence. Similar help for the town having a towny Mayor.
Incarcerations are iffy. Incarcerating a blue should be a last-ditch effort from a town-aligned Sheriff since it eliminates the benefits of that blue for the night. No DT checks, no medic protection, no Vigi hits that night (protecting a Vigi is, of course, slightly better than protecting a townie should the Vigi not want to hit that night and not have used his hit yet). Incarcerating a Mafia to lower KP is a great deal. Incarcerating a green can be good in certain circumstances (look at the last game Chuiu ran and Fishball's actions on day 4: http://www.teamliquid.net/forum/viewmessage.php?topic_id=89338¤tpage=69#1362 , or to protect a strong green player with no drawbacks). I think lowering Mafia KP is the best deal here but not necessarily something we'll have a chance to do.
One other thought--incarcerating a suspected Mafia when the number of Mafia is odd. IF that person is Mafia, it will lower KP that night. Medics know when they block hits, and Veterans know when they take a hit. From this we can essentially get extra rolechecks in these circumstances by just adding up the Mafia kills. (Mafia could stack hits, but that's still a win for the town. We don't get proof that the incarcerated is red, but the Mafia gives up an extra KP so it's still a benefit to the town). If we try this, I think it's best for Vigis to stay cool that night and not hit anyone, since a Mafia could try to claim the Vigi hit to keep their incarcerated ally safe. Clue analysis could fix this, but I think it's best to avoid the potential confusion.
For the mafia, the targets to incarcerate are blues that they think will get medic protection, or (probably ideally) incarcerating suspected medics to avoid getting their hits blocked.
My initial thoughts here are that the role of Sheriff is more beneficial to the town and Mayor is more beneficial to the Mafia.
Following from my above discussion of Sheriff and KP, if a medic blocks a hit we have a nearly-confirmed towny since the Mafia cannot hit themselves (not that I'd think they'd want to anyway). Having a confirmed townie to act as a leader is very powerful. Similarly, a Vigi hit can confirm a towny if that Vigi hits Mafia and can prove the clues point to himself.