Husband lighting Shabbos candles the Shabbos after birth:[1]

Wife is home: It is an old and ancient custom for the husband to light the Shabbos candles and say the blessing on the first Shabbos after birth.[2] [This applies even if the wife has already come home, nevertheless, the husband lights the Shabbos candles on the first Shabbos after birth. Practically, it is no longer customary for the husband to light the Shabbos candles on the first Shabbos after birth if his wife is already home. Rather, his wife is to light the candles as usual.[3] If necessary, then the candles are to be brought to her bed for her to light.[4]

Wife is not home:[5] If the wife has yet to return home, then the husband is to light the Shabbos candles with a blessing[6] as is the law throughout the year whenever one’s wife is away from home.[7] When the husband lights the candles, he should light two candles, and is not required to light the same amount as his wife.[8] [If there are other members of the house that will be lighting anyways, such as daughters, then from the letter of the law the husband does not need to light candles if his daughters are already above the age of Bas Mitzvah.[9] Nonetheless, he should do so anyways and light candles in addition to his daughter, especially if they are below Bas Mitzvah.[10] In such a case he should preferably light the candles that rest on the table. Whenever the wife is away from home and both she and her husband will be each lighting candles in their respective areas, then they are both to intend to not fulfill the others obligation with their lighting.[11]]

[1] Admur 263:5 “It is customary after one’s wife gives birth for the husband to light the Shabbos candles and say the blessing on the first Shabbos after birth. However, on all other Shabbosos, she is to light the candles even if she is a Nidah.”; M”A 263:6; Mateh Moshe 423; Shelah Shabbos Ner Mitzvah; Sefer Minhagim Tirana Erev Shabbos 26; first custom brought in Kneses Hagedola 263:10; Kitzur SHU”A 75:5; Aruch Hashulchan 263:7; Tehila Ledavid 88:3; M”B 263:11; Kaf Hachaim 263:23; Beir Moshe 8:66; Az Nidbaru 11:2; Shraga Hameir 5:11; Shevach Habris 6:1; Piskeiy Teshuvos 263:10; Nitei Gavriel 66:2

Other customs: Some are accustomed to have the wife light candles as usual even on the Shabbos after birth. If necessary, the candles are brought to her room for her to light. [Kneses Hagedola 263:10 that so is a good custom] Others, however, argue that is not a good custom being that the candles are meant to be lit on Shabbos table, and not in her room near her bed. [Birkeiy Yosef, brought in Shaareiy Teshuvah 263:2 based on Rama and Rambam] Others negate this custom being that it is not valid for the candles to be lit in one area and then moved to another area. [Tehila Ledavid 88:3]

[2] The reason: Several reasons are recorded to explain this custom: 1) As the wife was very weak the week after birth and could not be expected to stand up and walk to the candles to light them by the Shabbos table. Accordingly, the husband lit the candles on this Shabbos. [Hagahos Rav Baruch Frankel on M”A ibid; Toras Shabbos 263:4; ] Some negate this reason, stating that the candles may simply be brought to the wife for her to light near her bed, and so indeed was the custom of some. [Kneses Hagedola 263:10 that so is a good custom] Others, however, argue that is not a good custom being that the candles are meant to be lit on Shabbos table, and not in her room near her bed. [Birkeiy Yosef 263:3, brought in Shaareiy Teshuvah 263:3 based on Rama and Rambam; Kaf Hahaim ibid] Others negate this custom being that it is not valid for the candles to be lit in one area and then moved to another area. [Tehila Ledavid 88:3] 2) Others explain that the reason for the wife not lighting candles is because she is not clean from the birth and is in a state of Nidah, and is flowing a large amount of blood. [Aruch Hashulchan 263:7; Toras Chaim Sofer 263:6; Darkei Halacha 263:6; See Shevach Habris 6 footnote 1] 3) Others explain that the reason is because the entire reason for why women are given the Mitzvah to light Shabbos candles is as a Tikkun for the sin of Adam and Eve, which Eve instigated, in which she extinguished the candle the world. Hence, when the mother has given birth to a new child and rekindled the light of the world then she no longer needs to light candles for this Tikkun. [Beir Moshe 8:66]

[3] Rebbe in name of Rebbe Rayatz that so ruled the Alter Rebbe to his granddaughter, the wife of the Tzemach Tzedek, and that so ruled the Tzemach Tzedek to his daughter-in-law, the wife of the Rebbe Maharash, brought in Koveitz Hagahos Lassidur Rabbeinu Hazakein p. 28; Beir Moshe 8:66; Az Nidbaru 11:2; Rav SZ”A in Otzer Habris Tinyana 2:5 footnote 2; Shevach Habris 6:1 footnote 2; Piskeiy Teshuvos 263:10; Nitei Gavriel ibid

The reason: Some say the reason is because today’s women are strong enough to get up from their beds by the first Shabbos after birth. [Az Nidbaru 11:2] Others write the reason is because this would cause the woman great pain if she cannot light her Shabbos candles. [Beir Moshe 8:66]

[4] Rebbe ibid; Kneses Hagedola 263:10; Derech Hachaim 50:16 that it should be lit on a table next to her bed and remained there for some time, and then later moved before the start of Shabbos to the area of the dining table; See Piskeiy Teshuvos 263 footnote 98; 263:35

[5] See Beir Moshe 8:67; SSH”K 43 footnote 46; Piskeiy Teshuvos 263:10; Nitei Gavriel 66:5

[6] Piskeiy Teshuvos 263:10; See Admur 264:14 that when a husband is away from home he fulfills his Mitzvah with his wife’s lighting at home, and Rebbe in name of Rebbe Rayatz in Koveitz Hagahos Lassidur Rabbeinu Hazakein p. 28 that certainly he is to light without a blessing. However, in truth our case is not the same, as here we are discussing that the wife is away from home, and the above exemption of a wife for her husband only applies when the wife is lighting in the home of the husband using the husband’s candles, as explicitly written in Admur ibid; However, see Piskeiy Teshuvos ibid and 263:25-5

[7] Whenever one’s wife is away from home for Shabbos, then the husband is obligated to light Shabbos candles with a blessing, if there is no one else in the home that will be lighting. [See Admur 263:5, 9; 11; Kitzur Dinei Shabbos 3:7; Piskeiy Teshuvos 263:10]

[8] Rebbe in name of Rebbe Rayatz in Koveitz Hagahos Lassidur Rabbeinu Hazakein p. 28, printed in Shulchan Menachem 2:45; See Divrei Shalom 3:73; Chayeh Halevi 1:34; Pnei Shabbos 3:4-48; Piskeiy Teshuvos 263:2; See Nitei Gavriel 66:4

[9] See Admur 263:9-10

[10] Beir Moshe 8:67; SSH”K 43 footnote 46; Piskeiy Teshuvos 263:10; See regarding if the lighting of a small child exempts the father: Admur 263:11 [even a gentile is valid Bedieved and certainly a child below Bas Mitzvah, however he is the one to say the blessing and accept Shabbos afterwards]; Gloss of Rav Akiva Eiger 263 [Gentile is invalid]; Beir Moshe ibid [No, and therefore he must light]; See article of Rav Yosef Zevin in Yagdil Torah T.Z. 6:50

[11] Piskeiy Teshuvos 263:10 and footnote 102