When Chanuka begins on Shabbos and one lit the Menora prior to Davening Mincha should he say Al HaNisim at Mincha for Erev Shabbos? (Sources)
This very question is dealt with in chapter 43 of the second volume of אלה הם מועדי by Rabbi Eliyahu Schlesinger of Gilo, Jerusalem.
He cites Shlomo Zalman Auerbach who holds that one does recite Al HaNisim in Mincha after lighting the Menorah. (See הליכות שלמה- מועדים,יז:ז) He explains that despite that it's still the 24th of Kislev; it's considered already Chanuka for one who lit. In other words, it's not that it becomes nighttime for him- after all he still is able to light Shabbos candles after the menorah- rather it become the time of הודאה (=acknowledging God for the miracles/victory/salvation of Chanuka) for him and he thus must recite Al HaNisim.
Rabbi Schlesinger then cites שלמי תודה on the similar question as to whether one recites Al HaNisim in Birchat HaMazon if he lights menorah and eats while it's still daytime. The author of that volume reports that he asked Rabbi Aryeh Leib Shteinman and Rabbi Nissim Karelitz and they didn't provide a conclusive answer. (ונשארו בצ"ע)
Rabbi Schlesinger cites another source (חוברת מבקשי תורה-חנוכה ג, עמוד עו) that quotes Rabbi Eliashiv who writes that one should not recite Al HaNisim during such a Mincha. The reason is:
"Lighting the menorah and the days of chanuka are separate concepts. There's a law and time regarding lighting the menorah, and there's a law when the day of Chanuka begins in regards to acknowledging and praise."
In other words, the obligation to recite Al HaNisim and Halel is separate from the obligation of lighting the menorah. One who is unable for whatever reason to light the menorah would still be obligated to recite Al HaNisim and Halel. There is no concept of adding to the day of Chanuka- as there is for Shabbat. (i.e. one can accept Shabbat early- in the late afternoon after one and a quarter hours before sundown- and it will be Shabbat for him. However, one can't accept Chanuka early.)
Rabbi Schlesinger concludes that he finds Rabbi Eliashiv's reasoning more convincing and thus one does not have to recite Al HaNisim in such a situation. He adds that he also asked Rabbi Chaim Kanievsky who also agrees to this position.
Note: I have left out the overwhelming majority of the detail and logic of Rabbi Schlesinger's responsum. Please study his volume to get the full picture.
It is difficult to prove a negative, but I cannot find evidence that the lighting of the menora “brings in” Chanukah. It does not. (This could be seen to be supported by the need for the lights of the menora to last until 30 minutes after the stars come out on Friday night. (O Ch 679 (1) MB)). Therefore, there would seem to be no reason to say Al HaNisim at Mincha for Erev Shabbos.
I think your question arises from the custom that for women, the lighting of the Shabbos candles does normally bring in Shabbos, (O Ch 263 (10) MB) whereas for men there is no such custom (although it is better for a man to light with the condition in mind that he is not accepting the Shabbos at that time). FYI, MB points out that a woman should daven mincha before lighting and if she has not done that she cannot daven mincha afterwards. Instead she should daven maariv twice, using the second maariv to make up for the missed mincha.
Added later: I just found in Weekly Halacha by Rabbi Doniel Neustadt, the following (references to footnotes in italics):
Rabbi Neustadt does not mention saying Al HaNisim in mincha. The contradiction he mentions is that "Minchah "belongs" to Friday, while the Chanukah candles "belong" to Shabbos" and not to do with Al HaNisim.