There is a Minhag (custom) to light the Menorah every evening in the Bet Knesset, WITH a Beracha, to commemorate the miracle of the Menorah in the Bet HaMikdash. Many people are not aware that the custom also mandates lighting the Menorah again in the morning. This parallels the lighting of the Menorah in the Bet HaMikdash which would remain lit until the morning. This is especially appropriate according to Rambam who holds that the Menorah was relit each morning.
This custom is recorded by the Peri Megadim (670:1), Nahar Misrayim, Ohr Lesion (4:42:9). This custom applies only to a Bet Knesset, which corresponds to the Bet HaMikdash, not to a private house.
Just as with the regular Hanukah light, it is prohibited to derive personal benefit from the candles in the Bet Knesset. The Mishna Berura (Rav Yisrael Meir Kagan of Radin, 1839-1933), in Siman 674, holds that even the Shamush candle of the Bet Knesset is holy and may not be used for non-holy uses. He also brings down that the candles in the Bet Knesset should not be removed from their place for the first half hour.
SUMMARY
The custom is to also light the Menorah in the Bet Knesset every morning of Hanukah.
