How do you say "rock a baby to sleep" in Chinese?
Tentative answer: 搖
The question is not very clear to me. It seems that the request is for a word which unambiguously indicates that the mother is holding the baby in her arms and rocking it.  I don't think there is such a word in Chinese.  My first choice for the action of rocking a baby would be 搖, but like English 'rock' this is ambiguous as to how the rocking is done. Here are English and Chinese examples:

媽媽輕輕地搖著寶寶。
The mother gently rocked her baby.

In both the English and Chinese examples, the baby could be in the mother's arms, or in a cradle. I think the only way to clarify this in Chinese is to add a descriptive phrase. One possibility is to add the word 抱; this is the standard word for holding something in one's arms. If you use this, you will need to think a bit about how babies are held.  If you have the baby in front with its head resting against your chest, this is called 抱在懷裡. If you hold the baby with its head resting on your shoulder and one arm under its bottom, this is probably just 抱, so for example:

媽媽抱著寶寶輕輕地搖著.  The mother gently rocked her baby, (its head resting on her shoulder)
媽媽把抱寶寶抱在懷裡輕輕地搖著.  The mother cradled the baby in her arms, gently rocking it.

If you are trying to fit words to a song, try verb reduplication. 抱啊抱, 搖啊搖, etc. This how people actually talk to babies when they are trying to get them to sleep, or anyway this is how I did it.
A caution for interpreting
One thing to be careful about: English 'rock' refers to a semi-circular motion; Chinese 搖 is not limited to this, and sometimes just means 'to shake'. However, in English, to shake a baby is usually taken as crossing a very dangerous line, and might even be taken to mean that one is unfit to care for a child. If you ever interpret in a court case involving something like this, be careful!! You cannot take the word 搖 to automatically mean 'shake vigorously'!  You must explain to the court participants how the Chinese works and specifically ask in Chinese how the child was held and what degree of force was used; do not unfairly prejudice the judge or jury against the Chinese speaker.
My own feeling is that Chinese takes as conventional the sense that 搖 is appropriate for the baby.  Sometimes inexperienced children do it with too much enthusiasm, and the usual correction is 搖輕一點!