Twice in the past few hours, I've seen the idiom "don't give it the time of day". Now, I immediately knew and understood what the people using the phrase meant, but then I realized that I didn't know why that phrase means what it means. I Googled the phrase "time of day idiom" because I was particularly interested in the origin/etymology of the "time of day" part. I readily found the meaning (which I already knew), but was stymied as to its origin (which is what I wanted). Thus, I ask: what is the origin/etymology of the idiom?
An answer at answers.com quotes American Heritage Dictionary of Idioms to the effect that to not give someone the time of day means
Ignore someone, refuse to pay the slightest attention to someone, as in He's tried to be friendly but she won't give him the time of day. This expression, first recorded in 1864, alludes to refusing even to answer the question, "What time is it?"
By contrast, an answer at tywkiwdbi.blogspot.com says:
the expression goes far back beyond the time when people wore watches... In Shakespeare's day, the meaning was quite clear. "Good time of day" or "fair time of day" was a salutation just like "good morning" or "good evening"...We no longer greet people by saying "good time of day," but we still use the idea of giving such a greeting as a sign of favorable attention. In other words, refusing to give someone the time of day is thinking so little of him that you would not say hello to him on the street." (credit to Scribal Terror)
If tywkiwdbi and Scribal Terror are right, then the expression is rather older than the AHD Dictionary of Idioms said.
Edit: The BBC Learning English presentation “Not give someone the time of day” says:
Long ago, in Shakespeare's time, the phrase 'good time of day' was a greeting often used. These days we say 'good morning' [or 'good afternoon'] … So to say that you wouldn't give someone the time of day means you wouldn't want to greet them or say hello. So the saying means you refuse to give someone your attention.
As an example of the phrase used in Shakespeare's time, consider King Henry VI, part II, Act III, scene I, as Queen Margaret speaks:
Can you not see? or will ye not observe
The strangeness of his alter'd countenance?
We know the time since he was mild and affable,
But meet him now, and, be it in the morn,
When every one will give the time of day,
He knits his brow and shows an angry eye,
And passeth by with stiff unbowed knee,
Edit 2: Another example from Shakespeare (as pointed out by ΜετάEd's reference) is from King Richard III, Act I, scene III, when Buckingham says “Good time of day unto your royal grace!”. Note, Shakespeare is believed to have written both plays about 1591; he might or might not have put 16th-century speech into the mouths of 15th-century people. Henry VI was King of England and/or France at various times between 1422 and 1471, and Richard III from 1483 until his demise in 1485.
The phrase supposes that the smallest favour you can do for somebody is telling them what time it is when they ask.
So not giving them the time of day amounts to ignoring their tiny request, and by implication ignoring them.
In your example this has extended to ignoring some thing.