[Regarding Prophetic Practice, Innovation, and the Ruling of Recitation for the Soul of the Deceased]

The answer—and Allah is the One who grants success—is that Prophetic practice is that the worship of the legally responsible person is by what Allah, the Exalted, has permitted and legislated. Innovation is the opposite of that—exactly as He, the Exalted, said about a people who passed: "They have legislated for them in the religion what Allah has not permitted." [Ash-Shura:21]
Yes: reciting the Quran for the soul of the deceased is not an innovation. The evidence for that is that Allah, the Glorified and Exalted, has commanded His servants to seek means to Him through acts of obedience that bring one near to Him. He, the Exalted, said: "And seek the means to Him…" [Al-Ma'idah:35]. It is known that recitation of the Quran is among the greatest means to Allah’s mercy. For this reason Allah, the Exalted, began with recitation of the Book before prayer and almsgiving; He, the Exalted, said: "Indeed, those who recite the Book of Allah and establish prayer and spend from what We have provided them, secretly and openly, hope for a trade that will never perish." [Fatir:29]
So, if believers gather for condolences and recite the noble Book, seeking by its recitation that Allah have mercy on their deceased, overlook his evil deeds, and gather them and him in the abode of His mercy, then there is no innovation in that, for they have not transgressed the bounds of what Allah, the Exalted, has permitted and legislated. Seeking nearness to Allah, the Exalted, through acts of obedience is legislated; recitation of the Quran is legislated; and asking forgiveness and mercy for the believing deceased is legislated.
If it is said: recitation for the soul of the deceased at condolences did not come in the Book nor in the Sunnah; therefore it is an innovation.
We say: Allah, the Exalted, commanded seeking means and reciting the Quran in an absolute command that is not restricted to a time, a circumstance, or a purpose or request. That absolute wording indicates the legitimacy of seeking means by the Quran and by other acts of obedience for seeking forgiveness and mercy, or seeking healing and well-being, or seeking vengeance against an enemy, or seeking protection and safety, or other than that among religious, worldly, and otherworldly benefits.
What supports what we mentioned—that acts of obedience are means to attaining religious and worldly benefits—is His, the Exalted, saying: *"And whoever is mindful of Allah—He will make for him a way out, and provide for him from where he does not expect…" * [At-Talaq:2–3]; and His, the Exalted, saying: "And if they had remained steadfast upon the way, We would have given them abundant water to drink." [Al-Jinn:16]; and His, the Exalted, saying: "Worship Allah and be mindful of Him and obey me—He will forgive you of your sins and delay you until an appointed term…" [Nuh:3–4]; and His, the Exalted, saying: "So I said, 'Seek forgiveness of your Lord; indeed, He is ever Forgiving. He will send the sky upon you in continuing rain, and will increase you in wealth and children, and will make for you gardens, and will make for you rivers.'" [Nuh:10–12]; and His, the Exalted, saying: "And if the people of the towns had believed and been mindful of Allah, We would have opened upon them blessings from the heaven and the earth…" [Al-A'raf:96]; and His, the Exalted, saying: "And if they had upheld the Torah and the Gospel and what was revealed to them from their Lord, they would have eaten from above them and from beneath their feet…" [Al-Ma'idah:66]
And in the transmitted report: “Treat your sick with charity,” and “Seek provision to descend by charity,” and other than that from the Quran and the Sunnah.
This also indicates that reciting the Quran for the soul of the deceased is not an innovation: Allah, the Glorified and Exalted, has commanded prayer for the dead among the believers, and the funeral prayer is an intercession for the deceased with Allah, the Glorified and Exalted. In it there is recitation of the Opening Chapter (Al-Faatiha), and in it there is supplication—everything of that is for the sake of the deceased.
From here we know that reciting the Quran for the deceased and supplicating for him with forgiveness and mercy is not an innovation, neither at condolences nor elsewhere.
Yes: it is valid and permissible that a believer recite the Quran and gift the reward of his recitation to whomever he wills among the believers. What indicates that are matters:
1- There is no impediment imaginable to a believer’s soul being pleased to gift the reward of his recitation to whomever he wills among his believing brothers—whether living or dead—because the reward of his recitation is a right belonging to him from what he has earned. So he may dispose of it as he wills. No evidence has come—neither from the Book nor from the Sunnah—preventing that; thus there is no innovation in that. Recitation of the Quran is legislated; the reciter’s asking Allah, the Exalted, to make the reward of his recitation a gift to the soul of so-and-so is a good request and a dutiful supplication. A believer may supplicate Allah, the Exalted, with whatever he wills, and Allah, the Exalted, has promised response—except in asking for sin or severing kinship ties, as has come in transmitted reports.
2- It has come in a sound transmitted report that al-Hasan and al-Husayn and Ali ibn al-Husayn used to give the charity of breaking the fast at the end of Ramadan on behalf of all of their deceased forefathers (May Allah's blessings be upon them), and likewise others among the great figures of the Prophet’s Household (Peace be Upon Them).
3- It has come from the Prophet Muhammad (May Allah bless him and his family and grant them peace) that he sacrificed a ram on behalf of those in his community (May Allah bless him and his family and grant them peace) who did not offer a sacrifice.
4- It is established that pilgrimage on behalf of another is valid.
If it is said: Allah, the Exalted, has said: "And that there is not for man except what he strives for." [An-Najm:39]—which indicates that he benefits only from the reward of his own deeds, not from the reward of another’s deeds.
We say: the meaning of the verse is that a person does not obtain reward by way of entitlement except what was through his own striving. So a person is not entitled except to the reward of his own striving, not the striving of another. We mentioned “entitlement” in interpreting the verse because the preposition (al-lām) in the verse indicates it. The verse did not negate what the legally responsible person obtains of reward by way of favor.
This is like your saying: “So-and-so has nothing with me except one hundred as the wage of his work.” There is no impediment to your giving him two hundred—one hundred in return for what he is entitled to of wage, and one hundred as a gift from you, or through your mediation.
So in your giving the hired worker two hundred there is no contradiction and no inconsistency with your statement: “So-and-so has nothing with me except one hundred as the wage of his work.”

Source: Min Thimār al-ʿIlm wa al-Ḥikmah vol.3