For context: The product is for a niche audience, mostly free and is an opener for more products along similar lines.
A lot of founders think "When I launch, all my potential customers will see it that moment". This leads to delaying until it is "perfect" or waiting for the "perfect" moment.
In reality, when you release a few people see your product. And then a week later a few more. And then maybe someone who likes it shares it. You lose nothing by releasing now. The person who sees your product today and thinks "Unfinished and buggy" is not the person that buys your product in a month.
By delaying you lose valuable feedback. Maybe a key feature that takes 30 minutes to implement is missing. Maybe there's a bug that affects real users much more than you thought. You don't know until you have real users.
However if you are building a game or publishing on a specific platform launch dates do matter as they provide your most significant visitor boost from the platform. This is important because your future exposure might be permanently damaged because of a bad initial launch.
An example of this is avoid launching indie games on mass sales events or over christmas.
Launch early, and launch often.
If it were a physical product release, I would look at buying trends and correlate that with a shopping season that favors those trends. (for example; large household appliances are trending down during holiday season, but general electronics and toys are way up this time of the year).
You will likely not get press coverage of your launch. But if there’s an upcoming event you can tie into, it’s worth waiting a month or two to make it official. But you can still launch to beta testers in the meantime!
Life short, ship early and often!