Richard K. Guy refers to him as "F. Göbel" in his Unsolved Problems of Number Theory but that's as close as I can get.
For reference:
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1$\begingroup$ Nothing at Mactutor. $\endgroup$– SpencerNov 5, 2020 at 15:29
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2$\begingroup$ Göbel also appears to have been quite active in recreational mathematics as the designer of mathematical puzzles and contributor to the Dutch recreational mathematics journal The Dutch Cubist Club. $\endgroup$– nwrNov 5, 2020 at 18:46
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2$\begingroup$ For a number of years, Frits Göbel was also responsible for a column on recreational mathematics in the journal Euclides which is the main publication of the Dutch association of math teachers (Nederlandse Vereniging van Wiskundeleraren) from what I can gather. $\endgroup$– njuffaNov 6, 2020 at 0:07
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1$\begingroup$ Dr. Frits Göbel was part of the Faculty of Applied Mathematics at the University of Twente until 1998. $\endgroup$– njuffaNov 6, 2020 at 0:58
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The first thing I tried was searching for "Gobel" at the Mathematics Genealogy Project, which I figured wouldn't be successful because for something like this it's likely the person doesn't have a math Ph.D. (or any Ph.D.), and it's also something I would assume you tried. However, there was a single "F. Göbel" listed, Frits Göbel, and his thesis title also somewhat suggests that this might be the correct person. (A title such as "Enriched Categories and Quasi-uniform Spaces" would, for example, have had me thinking otherwise.)
Then I tried the google search "Göbel's sequence" + "Fritz" and got exactly one hit, this 25 September 1986 letter from Richard K. Guy to Neil J. A. Sloan. The sequence $1,$ $2,$ $3,$ $5,$ $10,$ $28,$ $154,$ $3520,\, \ldots$ at the bottom of the first page is identified with the name Fritz Göbel. The google search "Fritz Göbel" + "sequence" appears to confirm that this is the Göbel you want. I don't know whether he has a middle name or his birth year, but I believe he is still alive.
answered Nov 5, 2020 at 15:50
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$\begingroup$ Nice one! Didn't think of MGP. Looked in lots of other places. I see it appears to be "Frits" rather than "Fritz", which I will be careful to document. $\endgroup$ Nov 5, 2020 at 16:02
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1$\begingroup$ Oops, the Fritz spelling is a typo I made without thinking! It looks like this was a lucky accident on my part, since the Guy/Sloan letter uses "Fritz". $\endgroup$ Nov 5, 2020 at 16:04
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1$\begingroup$ Googling "Göbel's sequence" + "Frits" is actually quite fruitful. For a start, it's thrown up a work "Queueing Models Involving Buffers" which goes some way to finding out what he was all about. $\endgroup$ Nov 5, 2020 at 16:11
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3$\begingroup$ I found a paper from 2003 dedicated to Frits Göbel on the occasion of his 70th birthday, so his year of birth is likely 1933. This also jibes with the fact that he retired from the University of Twente in 1998, presumably at the age of 65. $\endgroup$– njuffaNov 6, 2020 at 1:26