Daisy is Doomguy's pet rabbit.
The rabbit's head is shown in the end picture of Episode 3: Inferno, first in a docile setting, resting in a sunny field, then in an instant turnaround, decapitated and its head impaled on a pike as evidence that the hellspawn have come to Earth. The rabbit is not specifically identified as Doomguy's pet at this time.
In the end picture of Episode 4: Thy Flesh Consumed, the rabbit's head reappears in Doomguy's hand. It is here that the game mentions "someone was gonna pay for what happened to Daisy, your pet rabbit". Due to the lack of any previous mention of this in the Episode 3 text, this would appear to be an example of retconning.
Daisy is most likely the reason Doomguy returns to Hell in Episode 4: Thy Flesh Consumed, as it shows in the end as Doomguy holds its head in his hand, possibly stating Doomguy avenged his pet rabbit, making Episode 4 a mission of revenge and is not totally focused on stopping the invasion of demons. Nothing is really known about Daisy, or how it came to be with Doomguy, as well as its past with Doomguy or its meaning or purpose in the game other than just a cinematic ending. However, many theories have come and gone over the years, many having to do with the bunny shape on the burning building. Daisy is more or less a mere ending for the Doom series and most likely means nothing in general.
Many state that the pike is none other than Doomguy's arm, however many discredit this as it makes little sense to as why he would do it and how the bunny got from the field to the front of the city. In truth, however, it is just a wooden spike. There are other theories stating similar things but none can be proven as not much is known about Daisy itself.