I've heard the theory that it's a place, but have never heard of a place with a similar name...
Adding to what jutky cited:
There is a parenthetical note in Rashi to Judg. 20:45* that after the civil war described there, in which most of the tribe of Benjamin was wiped out, some one hundred members of the tribe fled the Land of Israel and ended up "in the lands of Rome and Germany." Those who remained behind - including Eliyahu (or his ancestors) - came to be called "toshavim," residents (of the Land).
* The source given, מ"ע, is Megaleh Amukos by R' Nosson Nota Shapiro (17th century). This and many other such parenthetical notes first appeared in a 1699 edition of Tanach.
To add a little information to that which has already been presented, the ancient translations all interpret Tishbi as denoting his residence in a particular place. The precise name of that place, however, varies from version to version:
• Targum Yonatan: אליהו דמתושב מתותבי גלעד ("Elijah from Toshav, of the residents of Gilead"). So too Rashi, Rabbi David Qimchi (Radak) and R' David Altschuler (Metzudot Tziyon) on 1 Kings 17:1;
• Syriac Peshitta: ܐܠܝܐ ܕܡܢ ܬܫܒܝ ܡܢ ܬܘܬܒܐ ܕܓܠܥܕ (= Aram. אליא דמן תשבי מן תותבא דגלעד; ie: "Elijah from Tishbi, of the residents of Gilead");
• Greek "Septuagint" (3 Kings 17:1): Ηλιου ό προφητης Θεσβιτης ό εκ Θεσβων της Γαλααδ ("Eliu the prophet, the Thesbite of Thesbae of Galaad"). Aside from the addition of προφητης ("the prophet"), the Greek translator has understood "residents" (תושבי) to be the name of the place from which Elijah came;
• The Latin Vulgate (also 3 Kings 17:1) is painfully literal, but differs from the Greek: Helias Thesbites de habitatoribus Galaad ("Helias the Thesbite, of the residents of Galaad"). Presumably, they are also interpreting Thesbite as a geographical marker, but it is unclear whether they understand the town to be named Toshav or Tishbi.
In accordance with the Syriac, the HALOT lexicon interprets "Tishbi" as being the name of the place. In favour of that suggestion, note that Tishbi is also the name of a town in Naphtali, in which Tobias lived in the apocryphal Tobit 1:2.
Here is some info about Tishbi.
There is written that it is either a name of city (which has not been identified) or another form of the word "תושב" (resident)
The wonderful sefer Ramot Gilad by Rav Eliezer Weisfish (unfortunately not on Hebrewbooks.org) which is an encyclopedic work on Eliyahu Hanovi brings that Josefus in Kadmoniyos Hayehudim (book 8, 319) writes that Eliyahu came from the place Teshev in Gilad. This might be the same town as the Toshav brought by an earlier correspondent from the Targum Yonasan. The sefer Seder Hayom (https://hebrewbooks.org/pdfpager.aspx?sits=1&req=41230&st=%u05EA%u05E9%u05D1%u05D9&_rnd=0.7797483451474362) suggests that he was called that because as someone who lives forever can truly be called a Toshav, rather than Ger which applies to everyone else who only live for a certain amount of years! The sefer "Yichusei Tanaim Ve'amaraim" (page 328 -Hebrewbooks.org only seem to have the introduction of this sefer) writes that Eliyahu caused people to dwell in a town in Gile'ad and as such was considered as if he built it. Based on Pirke DeRebbi Eliezer ch 47 (https://hebrewbooks.org/pdfpager.aspx?req=22744&st=&pgnum=180) he suggests that he was possibly called like that because he caused, and will cause, people to do tshuva. From Rabeinu Rokeach the words Hishbiso ((Ruth 3-14) and Taashiv (Eicho 3-63) which refer to the coming of Moshiach, in which of course Eliyahu is involved, have the letters (Ha)Tishbi referring to him. He quotes one more pshat from the Sefer shaar Hagilgulim (hakdama 32) but as I don't understand it properly I can't quote it.
The word "תשבי" Tishbite, begins with the root word "שב" and means "to return". Adding a yod to the end makes it plural and becomes the word "שבי" meaning "to dwell". With the preposition tav the word becomes "He shall return to dwell". This definition makes more sense in the context. The word Elijah is a compound of the words "El & Ya" meaning "My G-d is YHVH". The passage then reads "Elijah the Tishbite" or "G-d is YHVH, He shall return to dwell".