What's the downside to rebranding company to my name?

The downside is if you ever decide to sell the company, it's something out of your hands with your name on it. Jimmy Choo doesn't own his own brand anymore, neither does John McAfee of McAfee Antivirus. Simple conclusion - don't brand it to your name if you ever intend to sell the company.

You're a specific company. And you're considering specific brand name options. So why are we operating blind and discussing your decision in terms of generalizations ?!?!?

I feel like I'm picking out clothing for you, never having seen even a photo. Hope your pants are a 46-inch waist and 28-inch leg ...

Naming is what I do. Give me a call.

What will happen when your business become an aquisition candidate? Will you sell your name as the brandname? The personal name may become a gray point to your possible Buyer?

I will be glad to have a conversation with you n that issue.

Is the business new? So is not rebranding.
Changing my your company's name to your own can set you back a bit. This is because a name is intended to be a conveyor of trust, are you an authority in your industry? Local Market? If so then it should help boost sales. If people are more familiar with the current company name specially since you are a reseller (if you are) of third party solutions and your value is in implementing those tools, you could be best keeping the current business name. For example, an insurance agent working under farmers brand conveys that he is just an agent. While John Smith insurance could sway either way depending on context in which a consumer crosses your path. They could assume your products are your own, if they are not that (could) lead to loss in potential customers. In the financial industry this is not uncommon however and if there is no brand to start with your efforts are the same. You have to build trust, with that you build authority and with authority you get to claim your name as a brand and trustworthy service provider.

A brand is not a logo. If people could care less if you change the name then you have no brand just a business logo and tax alias. Become a well known expert in your local market as you build your name sakes service... :)

There are ways to become an expert and ways to leverage that online. Some of us here have done that for ourselves and or for others.

Well it depends on a variety of elements, but most importantly: what is your name?

What is your desired scale for your company?: You may already know that you want to stay very small ("solopreneur"); that you will always be the face of your brand; in that you will be the one doing this training and up selling products; and that you will never sell or pass on the business as a brand. But using your name does not just affect exit strategies: you may change your mind and decide to grow by including other trainers or partners who might feel more vested in a business that does not revolve around one person's name. The brightest counter examples I can think of are the Kahn (Academy) and Lynda(.com)—but note the twists: neither are pure personal names—and neither started out with the intention of becoming big brands in the training space.

Your name already has meaning and context: All personal names have their own social, cultural, ethnic, and sonic qualities any of which can help, limit, or even kill your ability to differentiate your brand successfully based on your name. Example: A independent fashion designer client of mine was sued by a global fashion label because her first name happened to coincide with theirs—they essentially forced her to use her last name twice in her brand, sandwiching her first name which had to be smaller that the the bracketing last names. And I was left with the task of visually rebranding her company with a mark that makes it seems as if none of these constraints ever existed. So, you're unlikely to get anywhere with this strategy if your legal name just happens to be Charles Schwab. Conversely, a very common name might never catch wider traction simply (beyond your WOM, 1-1 client base) because it is unmemorable or generic. This applies to regular company and product names too—even from top brands. Try searching Google for Apple's Numbers or Mail or Pages support resources online... and see how many irrelevant hits you get compared to searches for InDesign or PhotoShop. When the name is rather generic the other factors such as the visual brand language (yes, the logo matters too!) have to over compensate to establish and maintain differentiation. So the question put before you by others here (What is YOUR name?) is very relevant. Even if your have a great name you would be missing out on crafting a name specifically directed at your type of service and your tagline, service descriptors, urls, etc may also have to also overcompensate.

Tightly coupling your name to the brand can have other risks: If you experience personal lapse or failure that may have nothing to do with your expertise—your brand will be tarnished. This would apply to a DUI or even a nasty child custody dispute or the best / worst case: Martha Stewart.

These are all just possible cautions to be aware of (there are many others). Still, a brand based on your name is probably fine especially if your primary (money making) interactions are 1-1 and you plan to stay small. Ask yourself, is my training approach generic to my field or unique (search: Stanislavsky System or Method)? Then, if you are really good at what you do and personally trustworthy these values will also gradually build-up on your brand—unless your name is something like Jim Bezzler.

In that case, please don't call me unless you want a new name.

Any company should be built to be "autonomous", a machine able to, one day, produce wealth without the founder being there.

While branding your company after your own name, you're tying your company, which is a productive asset, to you and your work as an individual.

You're buying yourself a job instead of a business.

A bit of a transition phase - which based on what you say will be minor.
More importantly you will struggle to sell your business if you want to, one day.