The most common and most likely answer to this crossword clue is the 5 letter word STYLE
I've seen this crossword clue in The Times
S
T
Y
L
E
More clues leading to the result STYLE
- It bears a stigma, part of nasty lesion
- Design steps for the auditorium
- Reportedly, rustic steps in vogue
- Miss perhaps having characters in nasty lesson
- Problem for looker left out of fashion
- Some nasty lefty type
- Give name to eyesore around Lithuania's capital
- Elegance of ancient writing-instrument
- Reportedly steps over fence in dash
- Approach an opportunity to get your leg over, it’s said
Buddy explains!

The straight part
We know that "Give name to" is the straight/definition part of the clue. The straight part is often a synonym or definition, in this case give name to = style.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 STYLE mean?
STYLE noun- Senses relating to a thin, pointed object.
- (by extension from sense 1.1) A particular manner of expression in writing or speech, especially one regarded as good.
- A particular manner of creating, doing, or presenting something, especially a work of architecture or art.
- To design, fashion, make, or arrange in a certain way or form (style)
- To call or give a name or title to.
- To create for, or give to, someone a style, fashion, or image, particularly one which is regarded as attractive, tasteful, or trendy.
- To act in a way which seeks to show that one possesses style.