No - you are not stuck with it for life!
Back in the early '70s, I started up a little pet shop and I named it Junglehut. After a few years, I moved on and started a traveling reptile exhibit named The Living Jungle. I did that with a partner for a few years, and then got married, and spent one summer running an exhibit in the Smoky Mountains - we called it Creature Feature. We lost money on that one, lol! So we moved back to Florida, and finally started breeding reptiles in 1985. We first called it Farmus Diversified, and then later, Glades Herpetoculture. When we took on a partner and started a "regular" reptile shop, we shortened it to Glades Herp. Then, when we split away from it, we started Cornutopia / Blue Chameleon Ventures. Those last two names only came into existence since 1996, but my first named business was around 25 years prior. And the only name most people might remember, other than our present names, is Glades Herp, and that, only because they are still in business.
It is fairly important to stick with a name once the business has been around long enough, and has done enough business, to build up a business relationship with a lot of people and to build up lots of business "goodwill". If the name is relatively new and relatively unknown, then it is very easy to change a website, and especially easy if you have not already gone through all of the steps to register your business with the local and state authorities, it is not that big of a deal to change it. Even after you are established, you can introduce a new name, and slowly work into it. When my accountant told me to incorporate, I just went with Kathy Love, Inc., but kept the DBA Blue Chameleon Ventures (hubby's photo and Mad tour business) / Cornutopia. Most people are not even aware that there is any other name. Much better to change it now, when it is fairly new, than after it is well established.
Also, if you are under around 20 - 25 or so, you will likely find that your taste in names, or possibly your business goals, may change in time, forcing you to also change your business name as well.