Star Trek: The Next Generation was originally supposed to feature a friendship between Wesley Crusher and Tasha Yar, an idea that would have been a mistake given the age difference between the two characters. Wesley, the teenage son of Doctor Beverly Crusher, served as acting-Ensign onboard the Enterprise-D for the first four seasons of The Next Generation and appeared periodically in subsequent seasons after he left to join Starfleet Academy. Wesley became a staple member of the crew and provided a vessel for the audience to view the Enterprise-D and her adventures with a childlike wonder.
Tasha Yar, by contrast, was the ship's chief of security until she was killed near the end of TNG season 1 in the episode, "Skin of Evil". Tasha was born on a failed Federation colony where her childhood was marked by poverty and violence. In contrast to this, she saw Starfleet and the rest of the Federation as a paradise and looked up to the other senior officers on the Enterprise-D, especially Captain Picard, whom she considered a mentor. Despite her brief time on the show, Tasha Yar left a lasting impression on fans and is still considered a classic Next Generation character to this day.
From their backstories, Wesley and Tasha seem to have very little in common. However, the writer/director guide for Star Trek: The Next Generation reveals that creator Gene Roddenberry planned for the two to develop a friendship. In Tasha's character biography in the guide, Roddenberry explains that because of the circumstances of her childhood, Tasha was supposed to treat Wesley like the childhood friend she never had. It is unclear from Wesley's biography what his feelings about Tasha would have been, but given his trusting personality, it is likely that he would have welcomed her friendship readily.
It is also unclear why the idea of Wesley Crusher and Tasha Yar's friendship was ultimately scrapped, but when all factors are taken into account it is obvious that the relationship would have been a mistake. Friendships that involve two people of vastly different ages always have the potential to become uncomfortable to watch, especially when they also involve two people of the opposite sex. Wesley was 15 at the start of Star Trek: The Next Generation, while Tasha was supposed to be approaching 30. A friendship between the two of them would not only have infantilized Tasha's character but provided the possibility of romantic or sexual tension on Wesley's part, something that would have been vastly inappropriate and might have lead to some uncomfortable storylines.
Perhaps if this Star Trek relationship was more like that of a mentor and mentee, it might have worked better. Additionally, if Wesley had lost his mother instead of his father at an early age, a close relationship with an older female character would have made even more sense. Wesley might have seen Tasha as a mother-figure, and Tasha could have taken the opportunity to mentor Wesley in an effort to make sure he was provided with the childhood she never had. The fact that Wesley's mother was still very much alive, however, meant that he would be more likely to look up to a male member of the crew as a father-figure in the absence of his own father.
This did end up being the case, with Wesley having several father-figures among the crew. Geordi La Forge and Data in particular took Wesley under their wing when he was promoted to acting-Ensign and mentored Wesley's interest in science and engineering. Any possibility of a friendship between Wesley and Tasha Yar ended when Tasha died, although, in perhaps a nod to Gene Roddenberry's original idea, Tasha's holographic will expressed regret that she would never get to see Wesley grow up to become "an exceptional young man". Star Trek: The Next Generation did the right thing in not including their friendship, though, and leaving it out paved the way for more interesting storylines for both characters.