You mean like they do now? You mean that someone has zero initiative to better themselves should receive a pay increase?
Don't talk about mentally ill or disabled or those widowed.
Give me one reason why I should change my opinion on a grown adult making a living working 40 hours a week at a minimum wage job.
Because if the business they work for doesn't pay them at least enough to scrape by, they'll begin taking your money directly in the form of taxes. You'll have no choice but to subsidize their employer. The money those employers save by paying $7.25/hr vs., for example, $9.50/hr are, in essence, a government subsidy because in order to make up the difference the employee will get food stamps and other hand outs to get by. Don't like Burger King? Subway? Ed's Lawn Care? US Bank? Too bad, your tax dollars are subsidizing them.
On the other hand, if the business pays a sustainable wage (bear in mind, I'm not calling for $15/hr like other people are, only a modest bump to the current $7.25/hr to about $9/hr) you are no longer forced to subsidize that business as a single person with no extenuating circumstances should be able to work 40 hours/week and get by. Less people relying on the government, less of your tax money subsidizing businesses.
I can only assume from your avatar you don't want people living on welfare and you want to have a say via the free market which businesses receive your hard earned dollars. This would do exactly that.
EDIT:
Let me reiterate: I am NOT advocating for a minimum wage high enough where a single mother could sustain her and the baby, a disabled person could get by despite high medical bills, or other extenuating circumstances. I'm saying if a single person with no other needs besides a place to live and food on the table works 40 hours per week then minimum wage should be enough to just barely get by. That's all.
If someone decides to start a family while making minimum wage, has large medical bills, wastes their money, etc. that isn't on the employer to subsidize their lifestyle or bad luck. Only to provide a meager means for survival which, to any normal person, should push them to strive for better than just squeaking by sharing a 2 bedroom apartment with a friend, driving a 15 year old car, eating crap food most of the time, shopping for clothes at Goodwill, and being lucky to have a spare $20 at the end of the month.