If the admiral "truly" knows YOU...and I mean really, down deep, KNOWS you...then he could write a letter of recommendation (a personal one) for you. But if he's "just a family acquaintance" that doesn't really "know" you...I'd leave it out of the equation.
I've seen many letters from generals and admirals when I sat on my MOC's nomination board. Most were the same: "I recommend...blah blah blah..." which told us that mom or dad worked for this person and that was all. The two that stand out though...one from a four-star general in the USAF (his was hand-written) and one from a navy rear admiral upper half (two star) were almost identical: "I highly recommend "Jan" for nomination to the XX academy! I have known "Jan" since I stood in at his/her baptism/since he/she was born...I've know his/her mom/dad since we served together in XXX and YYY and ZZZ and we've served together many times since. Our children are close friends, our families are close friends, I've personally observed "Jan" as he/she has grown through childhood in to the young man/woman that asks for your nomination. I can personally speak to their character, etc...etc...etc..." These were deeply personal letters from someone that truly knew the candidate.
That kind of letter is gold; at least it was to our committee, I can't speak to others.
So unless your relationship is that close...