Here's the the stuff in the game that provides dilithium that are shared by factions.
Faction shared (Federation and KDF)
Investigate Officer Reports: Still gives 1440 daily, just have to play a little longer. (not active right now)
Academy event: 1500-3000 seems to be the average range people can get, but the wiki says up to 5000 with luck. Sometimes available twice a day.
Lore missions: 480 daily for getting the right answer. Wiki has lore section for shortcut.
Borg Invasion: 480 daily
Exploration mission daily: 1440 for 3 missions in a cluster. Diplomatic investigation fills one out fast.
Aiding Deferi: 1440 daily, unlocks after beating breen episodes. Rescue and Emancipation unlock too this way.
Rescue captives: 480, daily. all you have to really do is rescue the Deferi from a Breen Patrol. Defeating the Breen is optional for Feds, a must for klingons.
Emancipation: Freeing Deferi slaves from Breen. 480 daily.
Asteroid mining: variable outcome, but tends to be several hundred. Daily.
Nukara prime: 640 for zone wides, daily. Thinning the Hive requires 250 kills for it to count.
Traelus system repair: 480 daily, fighting at the end of mission.
Sh'mar Distress call: 480 daily.
Defense contract: 480 daily, bonus of deuterium reserves at the end. Computer core fragments drop for Salvage dispute.
Salvage Dispute: 1440, requires 10 computer core fragments. Repeatable.
Big Dig Daily fleet action: 1440
Breaking the planet: 1440
relocate colonists: 500
transport settlers: 500
grant passage to colonists: 500
most doff assignments give a small amount of dilithium as reward for completing it.
Federation has 7 PvP missons that reward 1440 (daily?), Klingons have 6 that grant 1440, plus numerous ones that reward 480.
As you can see, there are plenty of ways of gaining dilthium still in this game. All it takes is a little bit of time to get it. That's why we have the system like we do.
Borg Sector Incursions: 480 each once per day
Fleet Actions: 1440 each ONCE per day, 480 per every run
Ker'rat Dailies: 1440 each (there are 2 of them I think)
Any Number of others I may or may not have forgotten: 480 to 1440 each.
DOff Missions: 5-2000 depending on mission
List is too damn long. If you're having trouble getting dil as a klink, you need a new brain.
I mine the asteroid some if I'm in the DS9 area. If not I usually hit the "Duty Officers" and "Explore the Clusters." By the time I do those, and my regular missions. I don't have much time to hit the other missions to farm.