Not in a low-mid level game with feats
I believe that the Storm Aura and Storm Soul effects are slightly underpowered before Level 10 when compared to other barbarians. The Storm Aura effects activate when you use a bonus action, which conflicts with Polearm Master (PAM) and Great Weapon Master (GWM) - two excellent feats for barbarians. The Storm Soul passive effects grant resistances and minor abilities only to yourself at Level 6. If you instead chose a Bear Totem barbarian, you would have all of those resistances at once (albeit only while raging). It isn't until Level 10 that the resistances affect nearby party members. I believe that allowing a character to change their aura environment during a long rest would not imbalance the Storm Herald when compared to other barbarians.
I will use Level 6 as a comparison; any feats are taken at Level 4. Attacks are computed against an AC 16 enemy with +2 to DEX saves. Damage per round is averaged over 3 rounds (the bulk of a fight). In my experience, GWM's bonus attack triggers in 1/4 to 1/3 of rounds, so I will assume that it triggers in the third round.
- GWM Bear/Tundra: 27.8 DPR = (23.8 + 23.8 + 35.7) / 3 rounds
- GWM Sea: 30.0 DPR = (26.5 + 26.5 + 35.7) / 3 rounds
- GWM Zealot: 32.7 DPR = (28.7 + 28.7 + 40.6) / 3 rounds
Desert's damage is comparable to Sea - it deals slightly more damage and may hit multiple enemies, but is party-unfriendly. If the party knows the day before that they will be fighting enemies with resistance to fire or lightning damage, the Storm Herald gets a small boost. If not, nothing about the class changes.
Computing Tundra's usefulness is a little tricky. You are exchanging a small amount of damage for a small amount of survivability. I can't think of a particular scenario where being able to choose it the day before would be much more powerful that simply choosing it at level up.
Storm Soul outside of combat
You gain resistance to fire damage, and you don't suffer the effects
of extreme heat, as described in the Dungeon Master's Guide.
Desert's always-on resistance is nice if you happen to step on a fire-based trap (and fail your DEX save despite having advantage) or if a fire attack/spell hits you before you can rage (despite gaining advantage on Initiative checks at Level 7). According to the DMG, extreme heat only affects characters without sufficient water. If you happen to end up in such a situation, you can make a CON save (DC 5, +1 per hour) to avoid the effects. If this is ever a problem for you and your +6 (or more) to CON saves, your party is likely already dead.
You gain resistance to lightning damage, and you can breathe
underwater. You also gain a swimming speed of 30 feet.
The swimming speed and underwater breathing are cool. Let's assume you didn't have them, though. If the need to breathe underwater comes up suddenly, then you can stay under for 4+ minutes with 16 CON. If that is a problem, your party is in trouble long before you are. If, instead, the party had time to prepare, the druid or wizard can cast water breathing as a ritual over breakfast that morning. Movement underwater without a swim speed effectively cuts your speed in half. Thanks to the barbarian's Fast Movement, this isn't as big of a deal for you.
You gain resistance to cold damage, and you don't suffer the effects
of extreme cold, as described in the Dungeon Master's Guide.
According to the DMG, extreme cold only affects characters who don't have cold weather gear. If you happen to end up in such a situation, you can make a CON save (DC 10, +1 per hour) to avoid the effects. Similar to Desert, your party will be dead long before this is a problem for you.
What about Level 10?
Sharing a resistance (while you are raging) with party members near you is certainly a nice feature. (Just don't forget that your allies will be close enough to take - resisted - damage if you activate Desert). The usefulness of switching between them depends on your campaign. Do you know you will be fighting a red dragon tomorrow? Being able to switch to Desert is nice...if whoever is standing next to you, face-to-face with a dragon, can survive the dragon's multiattack as well as you can.
Does switching environments on a long rest provide a power boost to the character? Yes. Does that boost lift the Storm Herald above other barbarians? I don't believe it does.