What happens to a Summoner's Eidolon if it is Petrified? Does it return at the beginning of the next day as if it were killed, or must it be freed of its effect first?
A petrified character is considered unconscious, not dead. To restore the Eidolon back from being petrified, it would need to have the process reversed somehow. Destroying the petrified Eidolon utterly might enable the Summoner to call it back as if it had died.
According to the rules:
Eidolons are treated as summoned creatures, except that they are not sent back to their home plane until reduced to a number of negative hit points equal to or greater than their Constitution score.
The eidolon remains until dismissed by the summoner (a standard action). If the eidolon is sent back to its home plane due to death, it cannot be summoned again until the following day.
According to the rules on summoning (emphasis mine)...
Summoning: a summoning spell instantly brings a creature or object to a place you designate. When the spell ends or is dispelled, a summoned creature is instantly sent back to where it came from, but a summoned object is not sent back unless the spell description specifically indicates this. A summoned creature also goes away if it is killed or if its hit points drop to 0 or lower, but it is not really dead. It takes 24 hours for the creature to reform, during which time it can't be summoned again.
Because the summoned creature is being reformed on the current plane, I would argue that when an eidolon is petrified, it can be dismissed, and resummoned (incurring the normal one minute penalty to cast the ritual to summon it).
I recall there being some better explanation of how summoned creatures work, but I can't seem to find it at the moment.
Sadly this is not covered by the rules and Paizo has thoroughly refused to say anything on the various forum posts asking about persistent effects and eidolons. So all answers on this are opinons. The way they've explained eidolons is in a weird grey area between really being a persistent creature like an animal companion and being a completely ephemeral creature like a summon.
I personally rule that you can rid your eidolon of persistent effects by dismissing them for the full day, as if they had been killed. It seems to be a happy medium between having to limp around with a gimpy eidolon and continually pay extra money to get it clerical help, and letting such effects be couple-round inconveniences.
I can't recall the source of my information, and admittedly can't be bothered to hunt it down right now, but I can say for sure this is what happens. The summoner becomes unable to magically or naturally sleep or fall unconscious, but can still die. The eidolon can't be forcibly returned to it's home plane in this condition, even through banishment or dismissal. The summoner will gain fatigue and exhaustion as days pass, be unable to rest to recover spells per day, and lose all abilities requiring the eidolon. As the eidolon is remained forcibly summoned, the summoner also can't use their summon monster ability. It's nearly as detrimental to a summoner as it is to a player who GETS petrified.