Thought I'd list out the spells mystics can increase duration to 24 hours using the lingering spell epiphany by connection to see which connections benefit the most from this ability. hour/level spells don't really make much difference (actually worse than doubling duration after level 12), so they are excluded. I'm including concentration up to 1 minute/level spells, assuming the duration could be expended if concentration could be maintained. Your mileage (and your GM's) may vary.
Crusader: reflecting armor (still ends when dismissed), Death Ward, Resistant Aegis, Resistant Armor, greater
Devastator: Carnivorous
Empath: Detect thoughts, Zone of Truth, Clairaudience/Clairvoyance (remote stakeout!), Telepathy, True Seeing
Geneturge: detect augmentation, Spider Climb
Hive Mind: Share language, Status
Melophile: Clairaudience/Clairvoyance
Shadow: Shadow Body
Shaper: grease
Star Shaman: Darkvision
Warmonger: resistant armor, resistant aegis
Xenodruid: Fog Cloud
Empath can do some very interesting things with this epiphany. 24 hour remote stakeouts with clairvoyance, True seeing/telepathy all day. Detect thoughts could be good if concentration spells are allowed, and zone of truth could let you enforce a "no lying on the ship" policy by recasting every day.
The only other one jumping out at me is shadow for shadow body. Cast once and reap the benefits all day. Crusader isn't bad either, but there are other ways to get energy resistance.
That being said, there are some other options. There could be interesting uses for a day-long fog or grease puddle. Though the fog could still be blown away.
