Six figures for a rebrand seems low. A handful of Agilex 7 SoCs would cost around as much.
Extremely low. It's not the "coming up with the idea" that costs the money, it's everything else. Also, note that intel has used the lower case i in their logo since the 60s. My guess would be that using lowercase 'a' in the logo was declared by fiat from Intel to keep the association with Intel who will remain the majority owner. Maybe they debated it and spent a lot of money on consultants to tell them it as an OK idea, but, I think it's plenty likely that particular decision cost nothing.
The cost of the (re-)branding is developing several logo options within the requested theme, doing searches for legal issues or confusions with similar logos globally. Making sure the name or logo doesn't have negative or profane associations in other languages. Preparing several variants of the logo art, updating web pages and documentation, building slide deck templates for employees to use when they give presentations, developing artwork for use in advertising and marketing, legally registering the trademarks in all relevant jurisdictions, acquiring domain names, and so on. I'm relatively skeptical of the value of rebranding when the company structure isn't changing, but in this case they kind of had to do something, and most of the costs were not coming up with the logo.
By re-using the name Altera it probably cost a lot less on at least the legal side than a new name, but I would still guess closer to 10M dollars than 0.1 M. Still chump change compared to the $16B intel spent acquiring Altera less than 10 years ago, and probably less than the other reoganization costs of the spinoff, and less than the $4B Intel's stock fluctuates by on a daily basis (it's up 2% today on a ~$200B market cap)