Is there a Jewish tradition of naming your child using the first letter of the name of a recently deceased relative?
According to Rabbi J. David Bleich (in the first class of of his Ishut series in 2010, which alas YUtorah.org does not have. Here is the first class the next year, which hopefully includes him saying that link), the value of naming after a relative is Kabbalistic. He went on to say that this Kabbalistic benefit* only takes effect if in fact the name is identical. So if I name my child Moshe because it was my grandfather's name, or Chaim because it was my other grandfather's name, some sort of benefit goes to those relatives. If I name him Moshe Chaim, no such benefit goes to either of them.
However, Rabbi Bleich thinks the value of making family happy is greater than the value of Kabbalistic benefit. So, naming with an initial, or a gender-flipped equivalent, or a translation, or such does not achieve Kabbalistic benefit. It does achieve happiness of family (presumably), and as such is of value for that reason.
*(This benefit may be to the deceased who get some form of extra credit from their namesake's mitzvot over and above the credit the relative would ge for descendant's mitzvot. Alternatively, it might be that the baby gets extra doses of good midot from the deceased (which frankly could work without Kabbalah, and merely through psychology). Or it could be both. I am not a Kabbalist.)
My understanding is that there's a tradition (particularly in Ashkenazic families) to give a child the [Hebrew/Jewish] name of a deceased relative. This carries on the memory of the relative and also, through some metaphysical mechanism, causes the good deeds of the child to enhance the standing of the relative.
This is a custom, but not a binding precept of Jewish law, so there's a great deal of flexibility available. When the relative's name is not suitable for whatever reason, some people like to fulfill something like this custom by giving the child a name that is somehow related to the relative's name or to the relative. Using the same initial first letter is one version of this, and I've also seen people using a synonym of the name or a name that describes the relative's character.