"I give a watch to my mother."
Yes, that's why you use に.
If you used が then your mother would be the one giving the watch. が marks the subject of the sentence. The subject of the sentence is the one that does the verb: mother gives.
は is the topic marker. That's a whole discussion on its own, but I think the way in which you mean, は would be understood to be replacing が and it would be the very same problem.