As a summary, there is at least a trope of robots that (probably) unintentionally undermine their inventor's intentions, Douglas Adams was an admirer of Robert Sheckley, but as of yet I have not discovered that Sheckley was a direct influence on Douglas' "Deep Thought" as a specific detail.
There have been for a long time SF robots constructed to guide mankind the way - a non-satirical (at least not intentionally satirical) is Gernsback's Ralph 124C41+ (it's right in the name, "one to foresee for one"). The supercomputer run by monks in Arthur Clarke's "Nine Billion Names Of God" works on a similarly existential question as Adams' robot (and given that the stars go out at the end of the story, it might be that the universe disappared to be replaced by something even more bizarre and inexplicable). Stanislav Lem has created a whole universe by itself of robots (often created by other robots) that sabotage their creators by being whimsically literally minded (apparently the universe is mostly empty because Trurl was so exasperated by the stubbornness of his "can create everything" machine that he told it to create "nothing"). Lem is relevant, because Adams said in an interview that was aware and a fan of Lem.
Douglas was also aware, and an admirer, of Sheckley. There is some second hand evidence is a reddit thread here (with a post quoting an unreferenced interview where Douglas apparently said that everything he did had been previously done by Sheckley). A somewhat more direct source is a Wired interview with Writer Tom Gerencer, who gives this statement:
“In the course of becoming a fan of Douglas Adams, I read some
interview snippets with him; I think it was in Neil Gaiman’s book
Don’t Panic, which has some interviews with Douglas Adams in it. But
in there, he asks Douglas Adams about Robert Sheckley—this controversy
about ‘people say you’ve copied Robert Sheckley,’ and Douglas Adams is
like, ‘Well, I had never read his stuff, but when I did I was like,
“Wow, it’s really similar to my stuff.”‘
So naturally I looked this up in Gaiman's book. Not only are the two quotes above embellishments, also the timeline does not fit.
Robert Sheckley's name is mentioned three times (within an single question and answer) in the Gaiman-Adams interview. The actual q/a is
[adams] ... people kept saying, "if your write this stuff, you must
know the work of Robert Sheckley?"
[gaiman] I assumed you must have
read Sheckley's "Dimension of Miracles"
[adams] People kept saying
that, so I finally sat down and read it, and it was quite creepy. The
guy who constructed Earth... it was completely fortituous. Those are
coincidences, and after all there are only a small number of ideas. I
felt what I did was more akin to Sheckley than Vonnegut.
Gaiman's "Don't Panic" was first published in 1988, and the mention of Sheckley is somewhat casual. What is weird is that apparently Adams's was on record as a fan of Sheckely rather earlier than that.
There is an interview on Darker Matter, purporting to be transcribed from a penthouse interview done in 1979 (published on the website in 2007), where Adams says the following:
"I have done. The SF writers I like best are the ones who are funny,
and there are not many of them. Robert Sheckley. He's a very, very
funny writer. He's also a stylist. Very few science fiction writers
write English well. Robert Sheckley can.
"So Sheckley... and another guy I'm very fond of is Stanislav Lem, a
Polish writer who has been superbly translated into English. That's
doubly impressive because it's a very densely verbal style, with lots
of word play. Translating it into English must have been
extraordinarily difficult, and in many cases it's been very well done.
So we know that Adams was aware of Sheckley by the time he worked on the original HHGTTG radio play, even if the did not quite remember it in later interviews.
Wikipedia has a small paragraph on the similarities between HHGTTG and the (much earlier) Sheckley work it is usually compared to:
Similarity with The Hitchhiker's Guide to the Galaxy
Dimension of Miracles (1968) has been cited as similar to Douglas
Adams's The Hitchhiker's Guide to the Galaxy (1978). In an interview
for Neil Gaiman's book Don't Panic: The Official Hitchhiker's Guide to
the Galaxy Companion, Adams said he had not read anything by Sheckley
until after writing the Guide (the first volume in the series) and
having seen it printed, and later found some of the parallels between
the two works to be eerie, but after all a coincidence. Gaiman, in an
interview two decades later, and five years after Adams' death,
paraphrased Adams' comments saying that some of the resemblances had
been "disturbingly close."
So there is a probable influence. If this extends to the literary creation of "Deep Thought" is hard to tell. No positive proof, but it seems that Adams' memory is not necessarily something that can be relied upon, so there might have been something subconscious/unintentional going on.