If you want to maximize influence fast Troublesome Goblins is near-instant payout, Bring Criminal to Justice is good reward for a single easy mob, and Raging Trolls is great reward for 1-3 not-so easy mobs. Scout the Location is easy money.

If you want to maximize the influence / experience ratio Train the Soldiers / Arena option is instant payout without much xp. Caravan and refugee escort and supply delivery take longer but have good ratios.

Unrewarding missions are deliver the message, which is very low reward, Tribal Goblins, which you'll spend days trying to chase down for little reward, and bring food, which is a lot of work for very little reward. Not taking the latter mission can deplete entire regions, as every time you request a mission it's this one, and the offer, whether you accept or decline, wipes the food supply. Is as if you're the Horseman of Famine.

Finally there are near-impossible missions. These are even worse than unrewarding, as your companions will feel dishonored when you inevitably fail. Obviously there's Dispatch a Scout Party if you don't have the right troops. Bring medium-good quality scrap metal; you'll never get enough. "Kill 3 dozen One-Horned Warg Trainers" where the target is so rare you'd have a better chance finding an ice cream truck. "Kill 3 dozen Scout parties" if your map speed is too slow.

Hope this helps!