I'm going to go out on a limb and say that it's most likely that Dumbledore named Fawkes after Guy Fawkes, a religious terrorist whose 'guy' is annually burned on bonfires up and down the UK.
So, what do we know about Fawkes (the phoenix) and what do we know about Dumbledore?
- We know for a fact that Rowling named Fawkes after Guy Fawkes.
Q. What is Bonfire Night?
J.K. Rowling responds: Good question! We celebrate November 5th in
Britain every year. There was a plot to blow up the Houses of
Parliament. The ringleader of the plot was called Guy Fawkes (spot any
Harry Potter connection?!), and we burn him in effigy and set off
fireworks to celebrate not losing our government.
Transcript of J.K. Rowling's live interview on Scholastic.com, 16 October 2000
- We know that Fawkes has only had one careful owner, ergo Dumbledore likely named him.
Q. Who did Fawkes previously belong to and will he play a vital role in the next book?
JK Rowling: I am not going to answer about the role in the next
books, which probably gives you a big clue, and he has never been
owned by anyone but Dumbledore. You will notice that when Harry goes
back in the Pensieve in this book, Fawkes is never there, and no, I
am sorry, not in this book, I take that back. When Harry has
previously seen the study with a different headmaster he saw it with
Dippet and Fawkes was not there then. Fawkes is Dumbledore's
possession, not a Hogwarts possession
Cub Reporters Full Interview with J.K. Rowling Jul 18, 2005
His name was Frank Bryce, he lived in the village where Voldemort’s
father grew up, and he has not been seen since last August. You see, I
read the Muggle newspapers, unlike most of my Ministry friends.’
Harry Potter and the Goblet of Fire
- We also know that Dumbledore is sufficiently well read, in terms of Muggle authors, that he can freely quote the likes of Alexander Pope. Although Pope himself doesn't mention Fawkes, again, it's unlikely that you could read more than a few dozen novels about British society without coming across mentions of the Gunpowder Treason.
This quotation demonstrates that Albus Dumbledore was not only exceptionally well-read in Wizarding terms, but also that he was familiar with the writings of Muggle poet Alexander Pope. — JKR
The Tales of Beedle the Bard
To summarise. Dumbledore reads Muggle literature as well as reading Muggle newspapers avidly, where he would have come across hundreds of mentions of Guy Fawkes. JKR herself named Fawkes after Guy Fawkes. There seems no obvious reason to assume that Dumbledore would have named his flaming bird after a different Fawkes, although there are several that might fit the proverbial bill.