Hourly rate is the worst way to charge for your expertise. That's an employee mentality.

Charge per project and charge for outcome. People don't care what the hourly rate, they want to know if I pay you X, you're going to deliver the Y results I am looking for.

I charge between 2000 and 4000 dollars to develop a new corporate or product name. That covers three days (at the low end) up to 30 days (for much greater complexity and matching URL and other supportive writing).

I'm a branding professional and I have done this for the past +10 years. For a project like this you should definitely charge a fixed price and bill according to the client's perceived value of the income, that I imagine will be pretty high since we are talking about the branding of his business.
Usually these types of projects take longer than anticipated because with branding and naming there is a lot of back-and-forth, since this is something that clients like to be very involved with. Therefore, I suggest to calculate what you expect will take you to complete the project and add a contingency that could be a minimum of 20% more. I don't want to give you an exact number in US$ because I understand costs in other countries are very different so in % you should visualize it better

Here's a tip- when putting your Agreement/Proposal together never say you will DELIVER a name. Word it so they understand they are paying for your CONSULTING on that. (What if they don't like the names?) You are being paid for what you are doing- "Name Consulting". I also add a clause that explains if a name is derived in the course of this project by any other person, you still get compensated for the consulting i.e., time, and ideas you put into it. (I feel like I am writing this too late- anyway-- good luck)

A good rule of thumb is to go with hourly for large commercial projects and for large organizations (or combination of fix and hourly after certain threshold) while a fix flat rate accepted upfront for small businesses and startups.

You can calculate a decent (non-exagerated) hourly rate as if you were an employee somewhere doing that work and then add a multple of 1.5 or something for your business profit and charge that hour multiple as a flat fee for the project.

My name is Humberto Valle, I'm the co-founder of Unthink Digital Marketing - if you have a link with samples I'd really love to see your work. :)

I believe that the local business can be compared to a start-up. Let us look at the cost you might incur.
1. Pre-seed funding start-up

If you are funding the start-up yourself, either by moonlighting on it while working another job or using your own savings, a good general rule of thumb is to spend no more than 5–15% of your total start-up budget on a branding project. So, if you are investing $30,000 of your own savings in your start-up, plan on spending no more than $1,500–$4,500 of that on branding. If you are investing $0 in your start-up, well, 5–15% of $0 is still $0. Use 5–15% as a rough sense of the scale of investment, but the exact amount will vary depending on your business. For example, if you are in a deeply technical business-to-business industry where coming out of the gate with a strong brand matters less, you may not need to make a large investment in brand right away. Or if you are an experienced marketer, you may be able to save money by doing a lot of the basic positioning and messaging work yourself, and only outsourcing pieces like design or website development where you have less experience. Just remember that every hour you spend on these projects yourself means you cannot work on some other part of the business, so account for the opportunity cost of the time investment while making your calculations. If your start-up is in a category where “brand is everything,”, you may want to spend 15% of your start-up budget on branding.
2. Seed-funded start-up
Start-ups that have received over $100,000 in seed funding or are already seeing revenue more than $250,000 per year are clear candidates for investing in outside branding help. This will only help ensure you understand key branding concepts better and can plan your branding investments wisely. I still recommend spending no more than 5–15% of your total start-up budget on a branding project.
3. Venture-funded start-up
If your start-up is funded at the $1,000,000+ level and you have not made an outside investment in branding, you should seriously consider bringing in a professional firm to help you.
If you are starting a chain of health clubs, look for a firm that has helped launch other lifestyle or health-related businesses before. If you are developing a new SaaS business, look for a firm that is technology-oriented and has technology work examples on their website. Expect to spend roughly somewhere between $70,000–$200,000 on your branding, depending on deliverables.
Here are a few rough guidelines for what you could expect to pay for each element of a branding project with a typical branding firm:
1. Company name development project: $15,000–$75,000
2. Brand story/positioning/messaging: $35,000–$80,000
3. Logo / Visual Identity: $20,000–$75,000
4. Website: $30,000–$200,000
Besides if you do have any questions give me a call: https://clarity.fm/joy-brotonath