Thanks for the feedback. This caught my interest more than I originally expected, and I ended up thinking a lot about it and designing a number of cards. Here is the result: some of my thoughts on the mechanic, as well as the cards I have come up with. As elsewhere, I made these cards more as examples to illustrate my thoughts than as finished cards, so I have not given much thought to cost/balance/playability (except to the extent those are what I am thinking about).
By their nature, Dawn cards immediately draw comparison to Night cards. The concept is similar, adding a new phase and proving a set of cards to be played during that phase (which is also a time of day).
The biggest design issue with Night cards is that when they are played, vanilla bonuses (+Actions, +Cards, +Buys, +$; Nocturn didn't use Coffers/Villagers mats or VP tokens) are all but useless. You can't play Action cards, making +Actions useless, and you can't buy cards, so +Buys and +$ are no good. +Cards could, in theory, draw other Nights that you could play, but given the fact that (1) they would draw every other kind of card dead, and (2) when randomized with multiple sets, there's a strong likelihood of only having one Night, that doesn't work in practice.
Dawn cards are (not surprisingly), the opposite. Vanilla bonuses are always useful, as every other part of the turn is ahead of you. One thing they have in common with Night cards is that they do not spend an Action, so they are never terminal, and each Action given is a bonus:
(This was my first idea; lots of Dawn cards are going to be about waking people up, and to me that functions as an extra Action. the Rooster does nothing else, but self-Butchers when trashed. Xen3k's Rooster is a much better card.)
The big design challenge with Dawn cards is that if they are not in your opening hand. I think this is potentially a huge issue. The reason adding +1 Action to Moat while taking away its reaction increases its price by $3 (a huge jump, especially going from $2 to $5) while doing the same to Market Square only increases it by $1 (with $3 to $4 being possibly the least significant $1 increase of any price) is that the two cards you get from Moat, if Action cards, are drawn dead (without a villager/Villager), but the two cards you get from Laboratory are not. The Dawn cards from either Moat or Laboratory are drawn dead, and since you can't go back and change the price of those cards, Dawn cards (it seems to me) have the potential to be really unbalanced/hard to price/play, or possibly disruptive to the existing balances that exist in the game.
The easy answer is, price Dawn cards accordingly, and there's not nothing to that. Take, for example, this card, which does exactly what Festival does, but only if it's in the hand that you initially draw, otherwise it does nothing.
This works in theory, but what if you are able to draw cards during/before your Dawn phase? For example, what about this?
On it's own, a non-terminal +3 Cards that has to be played at the start of your turn might actually be somewhat balanced at $4, since it is hard to build an engine around a card that has to be in your starting hand. The problem, of course, is that it doesn't really have to be in your starting hand. If your PotD draws another PotD, it is still your Dawn phase, and you can chain them. Suddenly, you can fill your hand with non-terminal Actions (preferably disappearing money) and treasures, and play all of them.
This is why I initially was supportive of one Dawn card per turn. I felt it would allow them to be more powerful without the risk of chaining a bunch of them for too much power. I've become convinced that this is probably not the best solution. As Aquila pointed out, this probably ends up worsening the problem, because if you only have 1 copy of a Dawn card then your really need to get it in to your starting hand. One solution is to have little-no-drawing among Dawn cards (a solution I don't love). A solution I like better is to make it so the Dawn cards either (1) won't be drawn during the Action phase, or (2) are not useless if they are.
One way to do this is to combine an on-Buy topdecking with a one-shot effect, making something like an Event-Duration:
(This is apparently an actual job people used to have, presumably before alarm clocks.) The card continues the theme of waking people up, but frankly, I kind of hate it. That said, topdecking is a promising mechanic, as it can (if timed correctly), be used to put a Dawn into a starting hand from which it can be played.
Another option (which a lot of people in the contest figured out) is to have Dawn cards be multiple types. Again, there is a good comparison with Night cards. Of the 15 official Night cards (13 Kingdom, 2 non-Supply), 7 are Night-Durations (including Ghost), 1 is Action-Night, and only 7 (fewer than half the total) are plain Night cards (including Bat). Of the 6 Night-Duration Kingdom cards, 4 give a vanilla bonus at the start of the next turn (alleviating that design challenge). Combining Dawn with other types can open up a lot of possibilities, and there are a lot of options.
One choice (which I went for with my submission), is to create a Dawn-Night.
In terms of flavor/theme, a good fit for these are
crepuscular animals--that is, animals primarily active during twilight (dawn and dusk), as opposed to those which are nocturnal (active at night) or diurnal (active during the day). There are plenty of official cards that are animals to use for guidance/inspiration, and the Wikipedia article in the link has a long list of crepuscular animals. They can do similar things that vary slightly by time of day:
Another solid option is Dawn-Reaction. There are two obvious (imo) ways to use this. The reaction can provide an alternative use for the card...
...or topdeck the Dawn card to be used the following turn.
Or, it could let the player choose between those two.
Another approach (which spineflu's card uses) is to have the reaction play the Dawn at the start of the next turn. Personally, I don't love that, because I do feel that even with helpful Reactions, Dawns are still somewhat harder to play, so their effect should be slightly overpowered for their cost, while cards that can be played when they are discarded are slightly easier to play and therefore end up being underpowered at their cost (e.g. Village Green). (I note that spineflu addresses this by giving a bonus for playing the card from hand).
One type which I think has a wider array of design possibilities when combined with Dawn cards than with Night is Action (as multiple submissions recognized). When these cards are played as an Action card it uses an Action, but if played as a Dawn or Night it doesn't. The one official card (Werewolf) takes advantage of this by being a terminal draw, with a Night card that lets copies otherwise drawn dead be used (albeit for a relatively swingy Hex giving).
The interesting thing about Dawn-Action (or Action-Dawn) cards is that they don't necessarily need language that distinguishes a Dawn phase play from an Action phase play. They could have the same text for both, but because playing it as a Dawn does not cost an Action, how it functions can be radically differently (see the above discussion comparing Moat and Laboratory) depending on when it is played. This distinction can turn terminal Actions into cantrips/disappearing money/non-terminal drawing...
...or turn cantrips/disappearing money into villages.
While I'm sure there are good designs that specify different actions during Dawn and Action phases (including Nobody and alion8me's submissions), I think the potential for elegance/simplicity with these is hard to beat.
(As an aside, while I don't especially love the regular Dawn cards in terms of appearance, these Night-split half Dawn cards--Dawn-Duration and Dawn-Night [with pink at top and bottom] and the Dawn-Action [with pink in the instruction box]--are some of my favorite cards aesthetically).
Another option (used by Aquila), is to do a Dawn - Treasure (here I like the look of the half-and-half split better):
The concept no doubt has potential, but I think the overlap with the Action phase is much more obvious and has a more promise.
I came up with a Dawn-Reserve:
As I've explained
elsewhere, I'm interested in the idea of a daytaler and think it has potential to make a great card or mechanic. This isn't it. I think this is overpriced at $2, and maybe even at $1. I haven't thought about the implications of a Kingdom-card that costs $0. This one is probably not worth doing any more with.
I also made a Dawn-Victory:
I think it's kind of nice (although I don't like the color combo either way). I think there is potential to the combination as it seems considerably more palatable to have a Victory card (even if it's a hybrid) that is frequently drawn dead.
Ultimately, I think the mechanic has a ton of promise and many possibilities. I am now inclined to have the rule allow multiple Dawns per turn, and have a card balance like Night, with fewer than half the cards being of just that type (and maybe an even lower percentage here). Even with plain Dawn cards, there is almost always a way to find the balance you need:
(This might not be it).
And just in case anyone was wondering: