Throughout the movies, we consistently see the Skywalker line being told to "search your feelings". Is there any significance to its relatively frequent recurrence, or the fact that the line was always said to a Skywalker?
It seems like its just something Force users say. It makes sense when coming from Jedi and Sith. Here's an excerpt from the Wookiepedia article on the light side of the Force:
The light side of the Force was aligned with happiness, joy, love, and benevolence, or alternatively simply calmness, which some believed nurtured the light side and provided insight into its ethical uses.In order to achieve harmony with the light side of the Force, its practitioners would often meditate to clear themselves of emotion; particularly negative emotions such as aggression, anger, and hatred, since these were shown to open a Jedi to the possibility of acceptance of the dark side.
And the dark side:
[Darth Revan and Darth Bane] claimed that while the light side represented the compassion and tranquility of all beings of the galaxy, the dark side focused on individual passion and strength. They also mentioned that while the light side is about maintaining the status quo, the dark side is about change and evolution.
So searching one's feelings is applicable to both sides of the Force. For Jedi, it asks the Palawan to calm themselves, for Sith it asks the apprentice to find an emotion and use it to their advantage. It's short, easy to say, and easy to remember; no wonder Jedi and Sith use it so much.
Out of universe:
As I said, it's short, easy to say, and easy to remember. In addition, it communicates to viewers a lot about the Force and is a good line to be famous. No wonder the writers decided to use it so much!
Those that are sensitive to the force can often feel an internal guidance coming from within that may sometimes resemble a feeling. For example, Obi-wan senses a disaster occurred in Episode IV:
I felt a great disturbance in the Force
Another example, in Episode VII, Snoke says to Kylo Ren:
There has been an awakening. Have you felt it?
Referring to the force awakening within Rey, or the light side of the force. In both cases, this sense is referred to as something being felt, or a feeling.
It also refers to having a higher sense of intuition. Obi-wan says to Anakin in Episode III:
Use your feelings, Anakin something is out of place
Later Palpatine uses a similar line to confuse Anakin:
Search your feelings. You know, don't you?
Since Anakin was confused at that point, he wasn't sure which "feelings" to go by, the dark or the light. Yoda also says to Obi-wan in Episode III:
Use your feelings, Obi-Wan, and find him, you will.
Based on these examples, "Searching your feelings" refers to the higher level of intuition that Force sensitives have.