In pretty much every shul and yeshiva I've been in has lit Chanukah candles by shacharis, the same number of candles that were lit the night before.
What is the source for this custom, and why is it done? Presumably, it isn't so that guests can be yotzei , because the only obligation to light is at night.
The custom is not to exempt the homeless, it's for pirsumei nisa; while nobody would notice candles during the day if they're by someone's house, today the lighting of Chunnukah candles in shul is noticeable enough that, while there's certainly no obligation to do so, the custom developed to light there as well.
The Pri Megadim (Eishel Avraham 670:2), among others, make reference to a custom to light Chanukah candles in Shul during the daytime, but, not being an obligation, no berachah is recited (Sefer She'arim Metzuyanim BeHalachah 139:19)
Otzer Kol Minhagei Yeshurin & Nitei Gavriel Chanuka 40:18 mention a reason from the Binyan Shlomo 38 that this is based on the Rambam that when they when one lit the candles in the morning he is also doing the Mitzva if the candles extinguished. Nitei Gavriel also mentions in the name of Nehar Mitzrayim that it is done for Pirsumei Nisa. The Nehar Mitzraim goes on to say there is more Pirsumei Nisa by day than by night, as by day there is no need for the light. This also assits those who may have missed the lighting the previous night with knowing how many to light that evening.
See also Rivevos Efraim.
