There are two minimum wages. One is a minimum the government sets for all employees to be paid and the other is the market minimum in which must be paid to attract and keep employees working under the conditions presented.
The government minimum is largely static outside acts of government but the market minimum is largely moveable with unemployment. The easiest and best way to raise a minimum wage is to reduce unemployment and let the market for employees demand a higher wage. This is a problem however because the economy has crapped out so badly that fast food jobs are seen as careers now instead of first jobs and stepping stones on a path to bigger and better opportunities. The worst part of this problem is that instead of looking at the situation and saying we need more and better jobs, a lot of people are content outside of wanting to raise the pay for these menial jobs.
When I got my first legal job (other than mowing yards and crap), unemployment in my area was low. I started at minimum wage but quickly progressed above it. All I had to do was basically show up on time and put a bit of effort at doing the work. My second job, which hired me away from my first employer at a raise, I went the extra mile for them and did pretty much anything they asked to the best of my ability and my pay quickly reflected it too. I was making 13 dollars an hour rolling burritos and flipping steaks on a charbroiler in the mid 1990s. $13 an hour might not seem like much but according to an inflation calculator one dollar in 1995 had about the same purchasing power as $1.56 in 2016. So in contrast, think of it as being paid almost $20 per hour in today's world and the minimum wage at then was $4.25 an hour (it went up from $3.35 when I first started working a few years prior).
If employment was there, by necessity, the minimum wage an employer could pay to keep an employee would by higher than the minimum wage. Those fast food jobs would go back to first job experiences and as stepping stones for people to show they can show up to work and follow directions enough to be hired at someplace that pays more.