Is there a standard expression used to greet or comfort those who are celebrating an anniversary of mourning? (...and why is it said?)
Even if not, what should one say? (...and why?)
We should merit to live till the days of Moshiach
The most common expression I have heard - "May the neshama have an aliyah".
The concept is that while the physical body remains in the earth, the dead person's neshama (soul) should rise to Gan Eden, and this is based on the person's merits. See www.jewishanswers.org for details. (I'll edit the full link, later, B"N.)
In Britain (certainly in Scotland) many ashkenazim say "you should have a long life" (not said as some kind of irony when being said to the one commemorating the yahrzeit but said in recognition of the mitvza of kibud av va'aym (honoring one's parents) which is reflected in their observance of their p'tira (passing) ).
Some say in Yiddish
די נשמה זול האבען אן עלייה
May the neshama have an aliyah