Barring magical/other interference, alignment is a player choice.

Nothing in the rules forbids this. While it is not explicitly said that player can change alignment at will, Player's Handbook (page 122) describes alignment as a choice:

For many thinking creatures, alignment is a moral choice. Humans, dwarves, elves, and other humanoid races can choose whether to follow the paths of good or evil, law or chaos.

Effects of alignment and broader discussion

Alignment serves as guidance for players to roleplay their characters. As characters develop over the course of a campaign with new experiences their worldview might change, prompting the player to change alignment.

Since alignment is mostly roleplay based it is up to the DM to decide how much (if at all) is alignment is going to be foregrounded in the game.

I have changed the alignment of my paladin from Lawful to Chaotic when he started to distrust the authority of his order. It allowed me to support his character development and show it on the character sheet without changing how the character played mechanically and we didn't encounter any roadblocks.

I have been allowing alignment shifts in games I DM (after talking to the player about their reasoning) and it went smoothly so far.

Possible Pitfalls

There are magical items that require attunement by creatures of certain alignment (or give out bonuses according to it). I have not encountered this but it is possible for players to initiate this change (or even try to switch alignment back an forth to be able to use these items).

Such items include:

• Robe of the Archmagi
• Candle of Invocation
• Sword of Answering
• Moonblade

and others. I consider it an abuse of a DM ruling and am vary of it

Exceptions to consider

There are certain effects that make DM change PC alignment (with the possibility of taking control of the PC by the GM). Most notably Lycanthropy. In this case, just letting the player change alignment back at will clearly goes against the intent of the mechanic.

Also see related: Are there official guidelines for a DM to change the alignment of a player's character? The linked question talks about the DM initiating the alignment change, not a player, but it is still relevant.

I use alignment descriptively instead of prescriptively.

If a character is played as Good, then they are Good. If the player changes how they play, the character can become Evil. When the change reaches the character sheet is up to the DM, based on time and significant events.

Players who want that change should be encouraged to make such changes gradually, and should be rewarded with an interesting story. The player should make their goal of changing known to the DM, who should work to lay out situations and experiences that highlight the character's progression. Once the character's actions have been consistent for enough time then the change should reach the character sheet.

I have found that new players are often confused by alignment, and I usually tell them to leave it blank or to fill in 'Neutral'. If a player chooses an alignment (usually Good), I never force them to act on it, but instead use it as a guide. Several times when a player has surprised me by making a particularly Evil action I have asked what their character's alignment was, and informed them that their actions weren't typical of the alignment that they chose or consistent with their behavior so far. I let them know that they can still choose the Evil action, but if they continue to do so that their alignment will change. The end result is that while I as DM determine when the character statistic that is alignment changes, it is the player who made the decision to change their behavior.

I have not read anything in the source material or otherwise (except as quoted in @Deeps answer) that suggests a better or more official way for alignment to change in 5e.