I would like to know if there is some online source where Boardman's 1964 thesis is available or his Warwick mimeographed notes. This is because by what I've heard Boardman's construction has a more modern or categorical feel to it than Adams's account in his blue book and I would love to look at Boardman's construction with modern eyes, taking into account all we've learned about the $\infty$category of spectra in the last years.

9$\begingroup$ I think I have only ever seen these in the library at Warwick. You could try bugging the people who are there. $\endgroup$ – Neil Strickland Dec 9 '19 at 16:23

3$\begingroup$ His student Rainer Vogt has a monograph called Boardman's Stable Homotopy Category. These were notes based on 1969 lectures by Boardman, and "more or less cover the first four chapters of Boardman's thesis". $\endgroup$ – Tyrone Dec 10 '19 at 13:46

1$\begingroup$ @Tyrone Thank you. I managed to get this monograph and it gives a detailed account of (hopefully?) Boardman's construction of spectra and the smash product. It is a thoughtstimulating read  at least for my some ideas were still new. $\endgroup$ – Lennart Meier Dec 13 '19 at 7:02
Boardman's thesis was (re)published a year later as three separate booklets, and a PDF scan of all three booklets is available on my scans page:
J. M. Boardman. Stable homotopy theory.
University of Warwick, Coventry.
Chapters I–IV: 42 pages, November 1965.
Chapter V: Duality and Thom spectra, 66 pages, January 1966.
Chapter VI: Unoriented bordism and cobordism, 71 pages, July 1966.

1$\begingroup$ Thanks for making these available. A fitting tribute to the man. $\endgroup$ – Tim Campion♦ Apr 25 at 19:37
Here is a link to Duality and Thom Spectra, which is the fifth part of the notes by J. M. Boardman that were mimeographed at the University of Warwick in 1965–1966 and covered the material in his 1964 Ph.D. thesis from Cambridge University, On Stable Homotopy Theory and Some Applications.
Reinhard Schulz, who has posted this file on his web site explains that parts 14 of Boardman’s notes are discussed in the proceedings of the 1998 conference held in honor of Professor Boardman’s sixtieth birthday.

$\begingroup$ Thanks. Sadly I am at least as much interested in Parts 14 and there is also not very much in the Proceedings you mention about these parts. $\endgroup$ – Lennart Meier Dec 10 '19 at 10:16

1$\begingroup$ A listing of 15 libraries that have the thesis itself is at WorldCat. $\endgroup$ – Carlo Beenakker Dec 11 '19 at 20:08

1$\begingroup$ @CarloBeenakker: None of these 15 items are Boardman's PhD thesis. $\endgroup$ – Dmitri Pavlov Dec 11 '19 at 21:55

1$\begingroup$ @CarloBeenakker: No, this entry does not point to Boardman's thesis. He defended his thesis in 1964, not 1969. The title of his thesis is "On Stable Homotopy Theory and Some Applications", not "Stable Homotopy Theory". This entry points to his 1969 notes published by the Johns Hopkins University, which are different from his PhD thesis. $\endgroup$ – Dmitri Pavlov Dec 12 '19 at 18:53

1$\begingroup$ @DmitriPavlov  thank you for correcting me; it is still very confusing for me, because the library entry explicitly says: "Thesis (doctoral)University of Cambridge, 1969" $\endgroup$ – Carlo Beenakker Dec 12 '19 at 19:32
Boardman's Ph.D. thesis is available at the Cambridge University Library: https://idiscover.lib.cam.ac.uk/primoexplore/fulldisplay?docid=44CAM_ALMA21579046710003606&vid=44CAM_PROD
The library refuses to lend it out through an interlibrary loan, so it appears that the way to go is for somebody at Cambridge to physically visit the library and scan the thesis.