The two sentences convey essentially the same meaning, but "that" is explicit, while "where" merely implies strongly. This is because "that" is used to state an identity, while "where" states a location.
In the first version, the phrase "the place that I need to search" makes that place the object of the search. This is the meaning that I think you intend to convey: that you will investigate the place itself (identified as the living room later in the sentence) in an attempt to find the watch.
In the second version, the phrase "the place where I need to search" only makes that place the location in which a search will happen. Because the object of the verb "search" is often a place, the implication is that the location in which the search takes place will also be the object of the search, but that isn't necessarily true. This version of the sentence doesn't contradict the implication, so meaning is conveyed, but you could contradict that implication in a sentence like the following:
The last place where I need to search is in the living room; my dog sometimes hides small objects under the couch there.
This version of the sentence has a much narrower implication, namely that the object of the search will be the space under the couch, rather than the whole room in which the couch sits.
In the example you provide, I think you could use either version and be perfectly understood, but I would tend to prefer "that" for reduced ambiguity.