Mes. Sofrim quotes different minhagim as to whether Eicha is read at night or during the day. It seems odd that we have adopted the practice of reading it at night and not during the day, as every other megillah is read during the day. The only exception is Esther, and even there many Rishonim hold that the primary reading is the one done during the day and not the one at night. Why should Eicha be different?
I think the explanation can be found in a question posed by Rabbi Maroof in a comment to a post from last year. He asked whether Eicha should be read by the ba’al koreh while standing, as the halacha demands for any act of kriah b’tzibur, or whether it should be read sitting, the way kinos as recited. I heard it read last year in a local yeshiva from a klaf with a bracha and the ba’al koreh sat. Eicha is perhaps read this way not just because it is followed by kinos or said in the context of kinos, but because Eicha is itself a kinah. Eicha sets the tone for all that we say and do on Tishab b'Av night and during the day. “Bacho tivkeh ba’layla… Kumi roni ba’layla…” Kinah is done at night, when the darkness adds to the sense of psychological despair and tragedy, and therefore Eicha, our paradigm of kinah, is recited only at night.
I would suggest the same can be said of our kri'as haTorah on Tisha b’Av. Kri’as hatorah is a kiyum of talmud torah – why are we engaging in talmud torah on 9 Av, a day when learning is prohibited? Would it not be more fitting with the character of the day to omit leining entirely? The answer I think is that the kriah is not done as an act of talmud torah, but also as a kiyum of kinah. (Nafka minah l’dina: can someone who is no fasting be called for an aliya on 9 Av morning? See Mishnas Ya’avetz by Rav Betzalel Zolti.)
R’ Bunim of Peshischa commented that of all that we had before churban, nothing remains except for Torah – “ain lanu shiyur rak haTorah hazos.” While the words of Eicha and “Ki tolid banim…” are tragic, the very fact that we have words of Torah to guide us in our grief is itself an element of consolation.