I cannot think of any first names for such phrase.
We do have something similar, but it would be closer to "John Doe":
This is "Kovács 132. János".
"Kovács János" is probably the most common surname - first name combination (equivalent of "John Smith"). For an emphasis of insignificance, we sometimes add a serial number in the middle.* For some reason, the number that pops into my mind is "132", others may use this phrase differently.

*This formula was originally used in football teams. When two players in a team share the same name, they are mostly referred to as e.g. "Kovács 1. János" and "Kovács 2. János".

Apologies from anyone who happens to be called "Kovács János".

(In the brilliant Hungarian translation of "Winnie the Pooh", the little bear lives in a hut with a board nailed to the top. The name engraved on this board is also "Kovács János")

English term (edited): every Tom, Dick and Harry

anyone = akárki
average person = átlagember

John Smith or Mr. Smith:
Gipsz Jakab (A Hungarian name used in examples.)

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Note added at 1 hr 8 mins (2005-10-31 17:38:43 GMT)
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The difference between "Gipsz Jakab" and "Kovács János":
There are many Hungarians with the name "Kovács János", while "Gipsz Jakab" is not so common in real life, but a typical Joker name in examples.

This combination was used to express the exact sentiment of Tom Dick and Harry by the Hungarian poet of the 19th century, János Arany [...amit ne értsen János, vagy Mihály...= that should not be understood by John or Michael, in Vojtina Ars Poeticája=The Ars Poetica of Vojtina - assuming that my memory is correct.) Now tiems have changed and the cited expression has never become used by others. Possibly the poet of today would put it differently.

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Note added at 4 hrs 16 mins (2005-10-31 20:47:15 GMT)
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In a very scarcely known song of the Chilean Quilapayun group (that was translated around 1970, if I remmember correctly, by P. Dabasi), the Spanish names Juan i Jose (and a third that I can't now remember) were replaced by János, Tamás és Kati. The meaning however here does not carry the somewhat belittleing if not ouright unpleasant overtones of Tom, Dick and Harry. Rather, they are replesentatives of the People. Notable, however that, touched by the times, a woman's name was included.

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Note added at 4 hrs 17 mins (2005-10-31 20:47:35 GMT)
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In a very scarcely known song of the Chilean Quilapayun group (that was translated around 1970, if I remmember correctly, by P. Dabasi), the Spanish names Juan i Jose (and a third that I can't now remember) were replaced by János, Tamás és Kati. The meaning however here does not carry the somewhat belittleing if not ouright unpleasant overtones of Tom, Dick and Harry. Rather, they are replesentatives of the People. Notable, however that, touched by the times, a woman's name was included.

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Note added at 7 hrs 11 mins (2005-10-31 23:42:02 GMT)
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Only for the sake of accuracy: the cited passage is in a different poem by Arany, titled "Vojtina Levelei Öccséhez, I" (Vojtina's Letters to his Cousin, I). As far as the intended meaning, there is no question that the citation is correct. It is a tongue-in-cheek instruction for the aspiring poet to choose words in his poems that won't be understood by Tom, Dick and Harry, or even by himself. (Te mindig olyan szót válassz, csinálj,/ amit ne értsen János, vagy Mihály;/ Legjobb, ha tenmagad sem érted azt..."). Hopefully this clarifies it for Csaba.

"Átlag János" = "John Average"
also used in this form: "átlagjános",
which is pronounced very similar to "általános" (=general, common).

...just for fun :)
This is a fancy name - the name of the first Hungarian lacquer maker appearing in a caricature of Eötvös Károly written by Karinthy Frigyes. Back translated the name sounds like "Gabriel Whothehellwas".
This name isn't part of the standard language, but may be a good choice, if somebody wants to shed light upon the exaggeration of a dead person's importance.