have found very little about the name Dzeidzeikowski in POland. I would be interested to simply know if there is such a name!
With probability close to zero - there is no Polish surname starting with DZEI. There is no such phoneme in Polish language.
However, there is a big chance that the first four letters of the surname have been rearranged and it should be spelled DZIE instead. The phonemes DZ, DZI and DZIE are somewhat different, but quite typical for Polish. There is a nasal variation of DZIE, spelled DZIĘ, which can be approximated as DZIEN and pronounced in English roughly as DJEN. Both DZIĘ- and DZIEN- could form the beginning of your ancestor's surname.
There is the same problem with the second syllable of his surname. It cannot be DZEI, but it could be DZIE, or even simpler DZI. This way we could convert the severely corrupted DZEI-DZEI-KOWSKI to either DZIĘ-DZI-KOWSKI or DZIEN-DZI-KOWSKI. Both sound more or less the same and have the same meaning, stemming from a dialectal verb "DZIĘDZIEĆ", which describes singing of a lark.
Those names are not terribly popular, but legal in Polish. Google shows 4550 results for Dziendzikowski, but 0 results for Dzeidzeikowski - not counting your posts on this forum and on genealogy.com.
The bad news is that the original name could have been corrupted in many ways. It could be originally spelled DZIDZIKOWSKI, DZIEDZIKOWSKI or even DZIECIAKOWSKI (from the noun "dzieciak", a child).