The most common and most likely answer to this crossword clue is the 8 letter word LACRYMAL
I've seen this crossword clue in Financial Times
L
A
C
R
Y
M
A
L
Buddy explains!

The straight part
We know that "weepy" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case weepy = lacrymal.The cryptic/wordplay part
In cryptic crossword there's often an wordplay/cryptic part as well.This part involves a more complex hint, such as an anagram, homophone, hidden word, etc.
Sadly, We don't have an explanation for this specific crossword clue yet.
What does LACRYMAL mean?
LACRYMAL noun- A lachrymatory (vase intended for collecting tears)
- (in the plural) Lachrymal feelings or organs
- One of the bones of the face, the os unguis, or nail-bone, in humans.
- Of or relating to tears or the tear-secreting organs.