Eladrin in 3.5e are completely different creatures from those called that in 5e, celestials rather than elves, exemplars of chaotic-goodness (in the same way that demons exemplify chaotic-evil, etc). They don’t have any particular ties to the seasons. They also don’t trance (or sleep). Some are technically playable, but it’s largely an exercise in frustration thanks to the effective character level rules.
In other words, you probably don’t want “eladrin” for this. Instead, you probably want the killoren from Races of the Wild. Each morning, these fey chose a different aspect of nature to manifest, which not only affects their personality, but also gives them different abilities. These aren’t based on the seasons, and there’s only 3, and since you can choose, most killoren choose the same one each day (whichever works best for their overall build). So it’s not a perfect match. But it certainly is close.
Moreover, whatever you play, eladrin or killoren or otherwise, yes, you can do this. Just role-play the character differently each day. There is nothing in the rules that stops you from doing that. Even with the killoren getting to choose, nothing in the rules stops you from “choosing” by assigning numbers to the aspects and rolling 1d3 each morning.
A few considerations that are worth discussing with your table:
Are you the only one of your kind who acts like this? If yes, why? If no, that needs to be discussed with your DM because all the others of your kind are up to them, not you.
Is this going to bother anyone at the table? Consistency in role-playing is generally desirable; being inconsistent can be disruptive. Figuring out where the line between interesting and annoying is—for your table in particular—is important (in general, but especially when you’re doing something unexpected).
If it has any mechanical impact (e.g. a killoren randomly choosing their aspect), is that weakness a problem? Killoren is far from the most powerful race out there, and not being able to synergize with an aspect you can count on might make them too weak. One could imagine treating this randomization as a Flaw, à la the Unearthed Arcana rules, but that would depend on the campaign (and whether or not you’re even using flaws).
