Gn39-02:03a forms a Biblical paragraph with a theme-detail-theme structure:
• Theme: And the Lord was with Joseph, and he was a successful man; and he was in the house of his master the Egyptian.
• Detail: And his master saw that the Lord was with him,
• Theme: and that the Lord made all that he did successful

Rashi interprets the details of this paragraph, the Lord was wish him as illustrative of the theme of this paragraph, Joseph was successful. Hence Rashi interprets the paragraph to mean that Joseph was a successful God-fearing person.

Rashi takes this concept of a successful God fearing person and gives typical characteristics of the successful God fearing person: Such a person will for example attribute successes to God, such a person will frequently say Thank God etc. Hence the Rashi comment: God's name was frequently mentioned by Joseph.

It is important to understand what Rashi is, and is not, saying:
• Rashi is not identifying the verse God was with him to only mean that God's name was frequently mentioned.
• Rather Rashi should be perceived as painting a picture of the prototype successful God fearing man. For example, suppose a movie or TV maker wanted to introduce a successful God fearing man: What would that person look like when he first appears? One approach would be to introduce a character that mentions God alot (...this success is due to God....)
• Rashi would not deny other characteristics of the successful God-fearing man: For example, prayer for success, giving to charity etc.

Advanced Rashi: Many Rashi-ists approach Rashi solely thru grammatical and word-meaning methods. They totally ignore stylistic methods. They also ignore Rashi styles of indicating examples of interpretation vs. wholistic interpretation. We could have interpreted the above Rashi, God's name was frequently on his tongue as implied by the Biblical text God was with him meaning God's name was with him. However instead we chose to interpret this Rashi stylistically: Joe's success was the success of a God-fearing person. Rashi then gave examples of characteristics of the God fearing successful individual. If we interpret Rashi this way we are then justified in broadly extending Rashi to interpret the passage as indicating other characterics of the God-fearing successful person (Such as prayer and strategy). This seems to me, to be the proper approach to these passages and Rashis.