Apparently if you adjust for both inflation and productivity growth it should be even more: $20. So $15 is some kind of compromise:
The economy has grown dramatically over the past 50 years, and workers are producing more from each hour of work, with productivity nearly doubling since the late 1960s. If the minimum wage had been raised at the same pace as productivity growth since the late 1960s, it would be over $20 an hour today.
I managed to track down the (brief) 2012 CEPR study on which that is based. I don't claim its calculations are correct, but the $15 figure is derived by splitting the productivity gain in half.
A final benchmark for the minimum wage is productivity growth. Figure
2 below compares growth in average labor productivity with the real
value of the minimum wage between the late 1940s and the end of the
last decade. Between the end of World War II and 1968, the minimum
wage tracked average productivity growth fairly closely. Since 1968,
however, productivity growth has far outpaced the minimum wage. If the
minimum wage had continued to move with average productivity after
1968, it would have reached $21.72 per hour in 2012 – a rate well above
the average production worker wage. If minimum-wage workers received
only half of the productivity gains over the period, the federal minimum
would be $15.34. Even if the minimum wage only grew at one-fourth the
rate of productivity, in 2012 it would be set at $12.25.
And the other tack is a cost of living calculation; I found a 2017 one for Illinois:
- Even in low-cost regions of the state, like Danville, East St. Louis and rural Illinois, workers
will soon need $15 an hour or more just to cover the basics. For example, by 2022 – the
date by which the legislature is proposing to raise the Illinois minimum wage to $15 – a
single worker in rural Illinois will need to earn $32,178 a year just to cover housing, food,
transportation and other basic costs, according to the Economic Policy Institute’s family
This translates to $15.47 an hour for a full-time worker. Similarly, in
Danville and East St. Louis, a single worker will need to earn about $15.55 an hour by 2022
to meet basic living costs.
- And in higher cost areas, single workers will need more. A single worker in Chicago with
no children will need $17.65 an hour by 2022 to afford the basics.
- Similarly, workers with children need even more than $15 an hour – even in the state’s
lowest cost regions. For example, in Rock Island – the lowest cost part of Illinois – for a
two-worker family with two kids, by 2022 each parent will need to earn at least $18.97 an
hour to afford a cheap apartment and cover simple living costs.
Given that the latter seems to be the data & reasons advanced in publications closer to the actual legislative proposals, I think this is probably the main argument, but I'm not sufficiently familiar with the matter to 100% conclude this.
As for the Federal level, I haven't found any living cost analyses indicating this precise figure. It seems it was more of a political thing, based on Bernie Sander's (2016) proposal:
“Democrats believe that the current minimum wage is a starvation wage and must be increased to a living wage,” the new plank reads. “We should raise the federal minimum wage to $15 an hour over time and index it, give all Americans the ability to join a union regardless of where they work, and create new ways for workers to have power in the economy so every worker can earn at least $15 an hour.”
Hillary Clinton's campaign added the words “over time” to original amendment, which was sponsored by former Ohio Sen. Nina Turner, a Sander ally.
The amendment was passed overwhelmingly by the 187-member Democratic Platform Committee.
Someone more familiar with the Democratic party may be able to find some internal documents with some kind of economic analysis, but I'm not sure they exist.
In a 2017 speech Sanders did bring up the "since 1968 ... the minimum wage lost 25% purchasing power" argument. So it seems somewhat similar to the OP's argument (which won't get to $15); later Sanders says something like "we know you can't make it on $12/hour". So it seems a standard-of-living argument, but it doesn't get anymore detailed than that, in that particular speech anyway.
The article in which that speech is embedded, also mentions that
Service Employees International Union launched the “Fight for $15” movement in 2012.
... from where Sanders eventually got his proposal. Interestingly, the year when that effort was launched by the service union coincides with the year for my 2nd quote, the brief CEPR paper. I don't know if this is causal (one way or the other), or just a coincidence.
Here's one of the earlier articles on this I could find, from Aug 2015:
In New York, a panel appointed by Governor Andrew Cuomo ruled that the state’s 180,000 fast-food workers should be paid a minimum of $15 an hour, in response to protests and pressure from Fight for $15.
Scott Courtney, the Fight for $15’s chief strategist, is helping to steer the movement through the Service Employees International Union, the union that has spent tens of millions of dollars underwriting the campaign. But Courtney has bigger ideas to expand on the notion of unionization.
If someone is inclined to dig into this further, they could look at that panel (did they write a report?) and/or how SEIU decided to back up the specific number. The latter might be a lot harder to find out... especially since there's a lot of time between 2012 and 2015.
An earlier press report from 2013 (and in a much more obscure publication) indicates a lot alternative slogans back then
The movement has no official name, though each city-level campaign has one: Fast Food Forward in New York, Raise Up MKE in Milwaukee, We Can’t Survive on $7.35 in St. Louis, Stand Up KC in Kansas City, and, in Chicago, Fight for 15 (which refers to a $15 minimum wage and has become the name most commonly used for the national campaign).
So at the level of slogans, "Fight for 15" won in a Darwinian selection. Also the movement did not really start asking for $15, but rather for $10:
Sidney (not his real name) says that when Action Now hired him in November 2011, it was to join a campaign to raise the minimum wage. For the first few weeks, organizers armed with postcards calling for a $10-an-hour state minimum wage prowled fast-food and retail joints in the Loop in downtown Chicago and gathered names, phone numbers, emails and home addresses to meet daily quotas.
On January 19, 2012, SEIU headquarters contributed $191,797 to Action Now, the first in a series of donations that would total more than $3 million by year’s end.
Apart from SEIU's involvement at this early state (merely 2-3 months after the fight for $10 (not $15) began, I wasn't able to find much, in particular how the jump from $10 to $15 in claims happened. Perhaps Denis' answer is correct on that.