Based on the clarification comment it seems to me that what you want to rebrand is your product and not the startup (company name). Rebranding a product is much easier than renaming your company and seems it would give you the benefit you are seeking. Your company name doesn't need to match your product name at all, it's a minor inconvenience while you are still small and in startup phase, you can worry about it when you are successful.
I recently rebranded my 7 y. o. site/product and here are the steps I took:
1. I looked for available domain names with keywords I wanted and unique phrase that didn't pull up any search results (you don't want a trademark problem). People say you don't need to use keywords anymore, but I am pretty sure it still plays a big role, so if you CAN try to use at least 1 word that describes your product. It will also be easier for your customers to remember than something generic or clever (.ly extensions). I found a suitable domain name on NameJet. Check for availability of the name on top social networks, you don't want a conflict with an account in the same space.
2. After I updated the site logo and colors, added redirect rules for all URLs for the old domain name, and put it on a new server I simply switched DNS to point to the new server. I picked a very low traffic time for my site to do the switch over.
3. Because my site was indexed by Google daily and redirects pass SEO juice, the new site started ranking for the same top keywords within 24 hours. I experienced NO loss of SEO rankings from the rebranding.
4. I chose to update social widgets to new URLs and I lost stats for Twitter and Facebook, but Google is smart enough to preserve votes for old URL. I had to rebuild stats from scratch for the other 2.
5. Once you have a site up on a new domain, you should switch over your twitter and facebook names over to the new name (based on availability you checked for before) and update descriptions/bios, if needed.
6. Write a blog post about rebranding. Send an email to existing customers and put a notification into site header about the site rebranding for at least 1 month so people know they came to the right place.
7. Optional. If your product name is descriptive, you might want to file for a trademark (under $500 via LegalZoom). A pending trademark application would also allow you to take over Twitter account or Facebook pages with the same name if the activity is in the same space as yours (especially if the account is inactive). Read up terms for both to understand the process.
TLDR: The hardest thing in rebranding a product/site is finding a good domain name that is available and name that isn't used by other businesses or by profiles on social networks. The actual switch over is just a matter of a long task list and right order.