I understand that this is a question on the Mishnah on 52a and 52b which states that the convicted person was sunk into Zevel.

1) I found, bs'd, in Toras Kohanim (the Beraisa of Chazal on Sefer Vayikra) in Parshas Kedoshim, that the Ra'avad in his commentary there explains this. This is on the verse in Vayikra 20:10, and in Toras Kohanim chapter 9, paragraph 11. The Ra'avad there (#11) writes that he does not know why the person was sunk into Zevel, because if the reason is so he should not move around and suffer, it would be better to use mud which is more honorable. However, the Ra'avad then suggests that the reason it was Zevel was so that when he opens his mouth he will die more quickly because of the bad smell of the Zevel.

We learn from the Ra'avad that Zevel was used in order to minimize the suffering of the convicted. (One also sees that it is possible to die from a bad smell, or at least that this can hasten the death of a dying person.) This way they fulfilled the commandment of, "And you shall love your neighbor as yourself," which Rabah bar Avuha explained (above 45a) includes the idea that one should "choose an easy death for him."

2) I found, bs'd, another early Rishon who writes about this in his commentary on the Toras Kohanim. Rabeinu Shimshon of Shantz (in his edition, it is in chapter 7, Parshata 10, paragraph 8, but I think the text of the Toras Kohanim is the same). He writes that the reason he was sunk into the Zevel is so that he should die quickly, because otherwise he could have been sunk into mud or into the land. Rabeinu Shimshon does not explain why he dies more quickly as a result of the Zevel, but at least we see from this that the Zevel was used in order to reduce suffering.

(We will see later on, bs'd, in the name of the Ra'avad (possibly a different Ra'avad than we saw above?) and in the name of the Tiferes Yisrael that Zevel was used to show how bad was the sin that the convicted committed. This does not agree with the Ra'avad and Rabeinu Shimshon on the Toras Kohanim, who say that it was used for the benefit of the offender.)

3) There is a different explanation in the Yad Ramah here. He writes that the reason why the offender is sunk in Zevel up to his knees is so that he cannot turn around and around. This is why he is sunk in Zevel and not in earth -- so that the witnesses who are carrying out the execution will not be dirtied with excrement. According to the Yad Ramah, the convicted was sunk in Zevel for the benefit of the executioners, not for his own benefit.

4) Other Commentaries explain that he was sunk in Zevel to prevent pain and disgrace.

a) The Shoshanim l'David on the Mishnayos (by Rav David Pardo 1718-1790) writes that it was necessary to sink him in the Zevel since, if he would turn over because of the pain, the executioners would not be able to pull the sheets around his collar forcibly and his death would be delayed. We do not want this to happen because of the Mitzvah of "you shall love your neighbor as yourself" cited in the Gemara on 45a as the reason that we should choose a less difficult death for him (see my first answer).

Again, we see that he was placed in the Zevel since this is the most effective way of preventing him from moving, and this makes the death less protacted and painful.

b) The Dina d'Chayei (volume 4, page 586, DH Hayu Meshakin, by Rav Chaim Benvenisti, 1603-1673) writes that he was sunk in the Zevel so that at the time of strangling his legs would not squirm which might lead to his limbs being dislocated and cause a disgrace to his body.

Dovid Bloom

Just to make one or two fairly obvious and practical points to conclude this topic:

It should be pointed out that the word "Zevel" usually refers to animal dung and is often mentioned in Chazal to signify animal fertilizer.

We noted that according to the Yad Ramah, Shoshanim l'David, and Dina d'Chayei, the convict was sunk into Zevel so that he would not be able to move. They understood that the convict will sink more deeply in Zevel than in mud and clay, which is why Zevel was chosen. The fact that he cannot move reduces the suffering.

We also noted that the Ra'avad and Rabeinu Shimshon of Shantz agree that Zevel was used because it reduced the suffering.

Dovid Bloom