I would suggest using your old domain, unless you are changing your complete, brand identity into a new entity.

1. Your domain name is a part of a hierarchy that Google reads, meaning that if you create a new domain, it will lose that ranking.

2. The new domain also means you will "tell" the search engines that you have a new website. The age of your website gives you relevancy according to Google and the older it is, the better you rank.

3. The activities that you have on your old website help with Google ranking and are indexed. This means you will either need to re-create these and build them back up with the new domain or may just lose some of the content and structure that you had.

Of course, there are several other factors that Google uses to help you rank, but these are all very important and take time to build.

If your website is being rebranded with completely different services, then it's worth making the change. Brand identity, quality and transparency are always first. If it is simply additions and changes, then it's best to keep the SEO value from the site you have. Continue to use what worked for you with the SEO and add into it as an improvement.

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I would create a sub domain. Use Newproduct.XName.com
That way it can be a branded website. This way you can keep the new name under new UI and not confuse the existing users.
By keeping under sub domain you can also pass the domain authority of main. if they are different product or services I would recommend Silo each other. if you need help how to solve I can offer free consultation on this platform. In Return for a good review as a teacher.

I would use the same domain, and unless the company is different buy a new domain and just re-direct everything to the updated old domain.

Happy to have a chat and explore whats best with more context

Old domain has a strong standing. Everyone knows it. Your new domain will take time to get recognition. So my advice is that you should use old domain to launch new product.

Perhaps it is wisdom to use both the names for about a year because the old domain name will bring prospective clients to you which you can easily convert into a proper sales. It is important to be visible and then wait for the invitation.

As per my personal experience I had www.angloindia.co.uk the name established with my clients but our partnership changed and we wanted to do business with a new domain name www.ProfitMentoring.com

When somebody clicked on www.angloindia.co.uk then the site automatically got transferred to www.ProfitMentoring.com and this way we got the maximum benefit of our old established name as well

It's impossible to answer your question without knowing some details about the "X" of your old domain and the differences between your old and new projects.

If X = BestHamburgersInLosAngeles.com. And your new project is sushi restaurant in Chicago, then you will need a new domain.