on a shoestring

Cheaply.

Not just "cheaply," but "as cheaply as possible" or "having very little money to begin with." The idea is that your shoe laces are almost the only resources you have. "South America on a Shoestring" would be a guidebook for people travelling on a very small budget, trying any way possible to save money--or to earn a bit of money so they can keep traveling. A cheap traveler might take local buses; a shoestring travelrer might walk or hitchhike.

You can also have a "shoestring budget" or work on a "shoestring project." That guidebook is for the "shoestring traveler."

The OED says:
2. A small or inadequate amount of money; a very little capital; a small margin. Chiefly in phr. on a shoe-string. colloq. (orig. U.S.).

[1882 Century Mag. Apr. 884/2 [He] could draw to a shoe-string, as the saying went, and obtain a tan-yard!]

However, someone on the American Dialect LISTserv uncovered a 1759 use in the Autobiogrpahy of John Adams
>She told the Behavior of the people, at the Tavern they were at in the Country about the Tea, before all the Monatiquot officers, shoe string fellows that never use Tea and would use it as [awkwardly?] as the Landlady did.

If I recall correctly, the entire metaphor is "hanging by a shoestring and a prayer", to signify the last moment before falling into the abyss.

Therefore, it first meant down to ones's last resources. Now it comes to mean 'barely getting by', or making your 'means fit the necessities'.

Hmm. I had never before heard "hanging by a shoestring and a prayer". (I assume you mean hanging on.) Google gives only 4 hits on that phrase. There are a lot more uses of "a shoestring and a prayer," but the "prayer just seems to be an intensifier. For example:

"We see businesses come into this area on a shoestring and a prayer," Major says, "and within a few months they're gone."
Think of the teenagers Mickey Rooney and Judy Garland putting together a really great show on a shoestring and a prayer.
On the day after Christmas 1988 with not much more than a shoestring and a prayer they moved to Atlanta GA

I wonder if the shoestring & prayer wasinfluenced by the phrase "coming in on a wing and a prayer," which means "you’re in a desperate situation and you’re relying on hope to see you through." It originated in a patriotic WWII song about pilots:

Comin’ in on a wing and a prayer
What a show, what a fight
Boys, we really hit our target for tonight
How we sing as we limp through the air
Look below, there’s our field over there
Though there’s one motor gone
We can still carry on
Comin’ in on a wing and a prayer.

I've heard, and used, the phrase "on a shoestring" or "shoestring budget" many times, in the sense cited above, "as cheaply as possible," where I've never heard "hanging by a shoestring and a prayer." Hanging by a thread, yes.

About 15 years ago, 4 of us did a 9 week tour of Europe, and at our first stop the rivet that held the pull handle on one of our largest suitcases broke. It was one of those bags with wheels only on one corner, so you pulled it with a strap, and without the rivet, it was impossible to pull it. Carrying it by the regular handle was possible but not comfortable.

So a repair was made tying the strap to the suitcase using a shoelace through the hole left by the defective rivet. It held for the entire trip, so we really DID travel through Europe on a shoestring!