Sleep is something that you need to manage personally, and even though companies like Google may have 'nap pods', they are by far a rarity and should not be considered the norm. It would likely be a bad idea to cite them as an example.
You can have 37 points on why your company should expend resources on a place for employees to nap, but the fact is every one of your reasons is mostly personal;
"Hectic schedules and stress"; Is your personal life stressful? Then it's personal. Is your work too stressful? Then they'll question if you can handle your position - nonetheless consider you for promotions.
"Caring for a toddler at night"; Intensely personal. It has nothing to do with your workplace. It was your decision to raise a child, your company won't accept child-rearing as an answer to sleeping at the office. If your child keeps you up at night, move its crib/bed away from your bedroom, or if you can't purchase $2 earplugs.
"Waking early for a long commute"; Again, personal. Look at moving or seeking more geographically appropriate work. Also, they might just say "Get to bed earlier" if you need to wake up early. It's not their fault you live far away. For reference, my office employs 3 people who live at least an hour away; none of us complain about sleep.
"Using too much caffeine, especially close to sleeping time"; Again, personal. Learn to manage your caffeine intake. If you're drinking coffee in the afternoon, stop. It's like asking your office to install dimmer lights because alcohol gives you a hangover.
"Waking up too soon and not being able to sleep quickly again etc"; Also personal. Additionally defeats your argument for sleeping quarters: If you can't fall asleep or wake up quickly, how will a 15-minute nap fix this?
It's just a bad idea
Not to be harsh, but your arguments will get you laughed out of the office. The fact is sleep is an intensely personal thing, and very few companies have the type of culture which will try to accommodate its employees on that personal a level. Simply put, if they aren't already doing it - it's highly unlikely they will ever start.
Additionally, even if you supplied the furniture, the company would need to set aside space, abide by hygiene laws & stipulations, and divert resources which could otherwise be spent elsewhere. The "Nap Area" could have easily housed another employee or storage area.
When it would be a good idea:
For the sake of completeness, there are some office situations that may benefit from 'nap pods';
Industries with overaggressive work hours are the biggest example, such as game development studios which may see employees work double-time during crunch periods; in this case on-site sleep to prevent overexertion may be needed.
Companies which may have remote offices; For example, research stations may be several hours away from 'civilization' or have unreliable transit. Situations like these routinely run shifts which span days to make the travel worth it, so sleeping quarters are included.
There are other valid examples, but generally speaking sleeping quarters are an extreme rarity, and usually only brought in when absolutely necessary. Companies like Google only do this because their employees are so highly prized they have to do incredible things to keep them; some of their engineers literally invented the fields they specialise in - that level of rarity in employees means their workplace scales to accommodate.
Other factors might be impacting your ability to work; check around the office and see if more cost-effective measures might benefit everyone;
- Is the lighting poor? If it is, then ask to have the lighting looked at, and see if others are affected by it.
- Is there air conditioning? Heat / humidity does actually make you more tired.
- Is there fresh air? Can you open windows? If you can't, consider bringing in some plants to the office if it's bright enough and you have permission.
- Is the work repetitive? Consider switching back-and-fourth between tasks.
- Are the chairs / furniture acceptable? Poorly designed chairs can cause you to slump, which in turn makes you tired.
- Can you convert your work area to a "standing workstation" (meaning you don't sit)
- Are you eating healthy food? See about bringing in small snacks or fruit and eating healthy throughout the day.
Lastly, and this one is personal:
- Have you been to a doctor lately? If you have severe sleep issues despite a HEALTHY lifestyle, you may have an issue like low blood pressure or nutrient deficiencies.