This search turned up very few hits. It seems people all over Japan utter the standard phrase “Hajimemashite” (初めまして) when meeting someone for the first time. That’s to be expected, I suppose. Why, even loquacious Americans can become rather stiff and uncomfortable around strangers. Dōzo yoroshiku onegaishimasu.
Tōhoku
Aomori
はじめやんして
Hajime yanshite
(In Hachinohe. Possibly used throughout the former Morioka-han (feudal domain), present-day Aomori and Iwate.)
Miyage
ど～も～
Dōmō
はじめっじゃんそか
Hajimejjansoka
(In Nishimorokata-gun, standard Japanese: はじめてでしょうか)
おんや、まんずまんず
Onya, manzumanzu
(In Sendai. Can also be used when you receive something.)
Akita
あら、初めでだんしな
Ara, hajime dedanshina
はずめでだんす
Hazume dedansu
(~だんす, dedansu is Akita-ben for desu)
あったことねぇやな、はじめてだよな
Atta koto nē yana. Hajimete dayona
(Southern Akita prefecture. A casual way of saying to a friend’s friend, “We haven’t met, have we? How d’ye do?)
Fukushima
ちわ～
Chiwā
(I suspect this is a contraction of konnichiwa)
Chūbu
Niigata
初だの～
(Not sure if this is read “Hatsu da noh” or “Haji da noh”, but it’s probably the latter.
どこさんさぁ
Dokosansā
(Sado-ben, spoken on Sado Island. Because it’s an island and everyone already knows everyone else rather than say, Nice to meet you, they ask where you’re from: どこの出身 -- Doko no shusshin?)
おみゃあさん、はじめてだなも
Omyāsan, hajimete danamo
(Nagoya-ben. Apparently only older women use this phrase nowadays)
Kansai
Ōsaka
まいど
Maido!
(Seems like they say this a lot in Ōsaka.)
Shikoku
Tokushima
おうたことないんちゃう
Ōta koto nain chau
(Standard Japanese: 今までに会ったことはないでしょうかーIma made atta koto ha nai deshōka?ーWe haven’t met before, have we?)
Okinawa
Okinawa
はじみてぃ、やいびーんやーさい
Hajimichi, yaibiin yāsai
(A casual, and rather long way of saying hajimemashite)