I would say some is nostalgia. Like fireball does d6 because fireball did d6 in prior editions:
AD&D 1E - Fireball throughout the ages
I am not sure if there is any strict logic, because when you look at monsters they cared mostly for the average damage (making average correct according to monster math).
What I can say is that normally ongoing damage is counted as doing 2 times damage (in average).
I can also tell you that dailies were assumed to do around 3dx damage at level 1 and 1dx more for each daily level up to 9dx at level 9 if it has no too big effect (except half damage in miss). This can be seen most easily in barbarian rage strike.
Strong effects often give 1 less dice or at least weaker than normal dice. Normal dice is d8 (2 upgrades and 2 downgrades possible).
In general encounter and daily powers can be seen as "X upgrades over at will" (his can be nicely seen with the psionic powers costing power points).
A level 1-7 encounter is 2 upgrades over an at will. 1 upgrade is normally 1 more damage dice. And 1 upgrade is a stronger effect (better rider). Instead of a dice upgrade you can also have 2 effect upgrades.
Encounter power upgrades can be seen nicely from the power points from psionics.
1,3,7 are 2 upgrades each (1 damage 1 rider normally)
13 is 1 more upgrade than 1 so 3 upgrades (2 damage 2 rider normally)
17 is 2 more upgrades from 3 (3 damage 1 rider or 2 and 2). This is important because on level 21 at will upgrades
21 the 2 power points I assume are because of the noemal upgrade to at wills. This is a bit strange, but I count them towards the at will upgrades
23 is 2 more upgrades than 7 so 4 (3 damage 1 rider normally). Important here is that at wills got also an upgrade at this level (in 1 damage dice), so this is 4 damage dice over original level 1 at will.
27 is 2 more upgrades than 13 so 5 (4 damage 1 rider) again over upgraded at will.
The 1 more upgrade at level 27 does balance out the level 17 which has 1 less upgrade (since at wills were not upgraded then).
We can also see the same structure in the essential classes. The encounter powers do 1 upgrade in effect, and 1 upgrade in damage per tier over at will/basic attack.
Daily 1 is 2 upgrades over encounter 1 or 4 over at will 1. (So 2 dice more than at will (1 than encounter) and stronger effect than an encounter normally).
For the barbarian rage strike we can only see the damage upgrades. I think the 2 effect upgrades are half damage on miss and "this is an alternative to rage" which normally has more upgrades in the encounter long effect instead of damage. (But because this "alternative" is not really an effect rage strike feels weak).
All higher level dailies are 1 more upgrade from the last. ( 4 at 1, 5at 2, 6 at 9, 7 at 15 usw.)
For lightning serpent I assume the 2d12 (13) to be used instead of 3d8 (13.5) to have slightly more variance, since the spell else is really consistent (slow always and 5 fixed ongoing damage). This would give the 5 damage dice (3 from 3d8 + 2 from 5 ongoing (ongoing counted twice 5 is 1 damage dice) + 2 effects (miss damage and miss effect).
Fireball the level 5 spell uses 4d6 for nostalgia over 3d8 (same damage less variance) and has 3 mechanic upgrades (burst size 2 times and half damage).
Prismaric beams the 2d6 are fire damage like fireball, also the other damage dice are all rather big (5 ongoings). It does 8 dice of damage (10 ongoing = 2×2, 5 ongoing = 1×2, 2d6) and has 1 effect (daze). The effect is slighly stronger than normal but the noemal half damage effect most dailies have is also stronger than a normal rider so it evens out. (7 damage dice upgrades)
I hope this helps a bit.
