Thimblerig is a surprisingly powerful code gate. Many Jinteki decks go for ICE like Aiki or Mind Game if they need a cheap code gate to pad their servers out with, but for me, I've settled on Thimblerig as my first choice. Here's why.
First off, as the reviews on its page mention, Quandary is a good ICE. Thimblerig is better than Quandary in every way other than its rez cost, and with a rez cost difference of only 1, it can do a passable imitation. Thus, putting a Thimblerig out early game forces the opponent to find some way to get through code gates.
In the early days of Netrunner, that would typically be a decoder. But times have changed, and runners have a wide variety of ways to get through ICE early on nowadays – if you try to use a code gate as a gear check nowadays, the Runner is probably more likely to bring out a Boomerang, Botulus, or Endurance than they are a Gordian Blade. These "pseudo-bypass" cards have the benefit that they don't care about the ICE type, and don't care about the ICE strength, so they're much better at stopping a rush early game than traditional breakers would be.
The genius of Thimblerig is that it's at least somewhat resistant to this sort of trick. Normally a scoring server will have multiple pieces of ICE on it; glaciers make giant scoring servers, and even rush decks usually stack multiple pieces of ICE of different types so that the runner has to find multiple breakers to get in. A common scenario is that you've rushed out an early agenda behind a gear-check barrier (say a Palisade or Ice Wall), but the Runner face-checked it and knows what ICE is there, and has their fracter out already. So you install your Thimblerig onto the same server, install and advance your second agenda, and the Runner puts a Boomerang on the unrezzed Thimblerig and runs in.
With almost any other code gate, the Runner would get in at this point, despite not having found their decoder yet. With Thimblerig, though, they get locked out; after they Boomerang their way through Thimblerig, you just swap it backwards in the same server, and now they have to get through it again – but Boomerang is single-use and they can't break the Thimblerig twice. This ability is well worth paying the extra 1 for – it comes up very frequently, and has won me a number of games. This isn't even necessarily limited to the early game; some Runners will run a nontraditional breaker suite, or have been pressed for economy the entire game and could never afford to install a permanent decoder, or don't have the MU for all the programs they want and are leaving out their decoder as the last important part, and I've even had a Thimblerig chain save the critical final agenda that would have put either of us up to seven points.
It's also worth noting that pseudo-bypass cards like Boomerang and Endurance, due to not caring about subtypes or strength, are equally good at breaking the vast majority of ICE. So a Thimblerig is doing pretty well in terms of "effort for you to rez vs. effort for them to break"; you could have spent twice or three times as much and had just as much impact on the Runner (or indeed less – most ICE can't force the Runner to encounter it a second time in the same run).
The other great advantage of Thimblerig is that it's a gear check that isn't dead even once the Runner has their decoder up and running. Apart from its usefulness in forcing a decoder install, it can do a separate job, tending to your servers and ensuring that all the ICE is in the optimal places. Often the best short-term placement for ICE is different from the best long-term placement, and Thimblerig bridges that gap, allowing you to (gradually, over time) fix placement decisions that were necessary at the time but suboptimal for the ongoing game.
Finally, Thimblerig has the strange property, unusual for gear checks, that (depending on what deck the opponent is running) it is actually occasionally taxing. The best you can normally hope for with a "cheap" piece of ICE (2 to rez) is that you're taxing the Runner 3 per run. For example, IP Block, which specialises in taxing centrals runs and is one of the best cheap pieces of ICE for the purpose, typically takes 3 to run through (less if the Runner has link) – and yet it is very bad as a gear check, being passable with no breakers at all. Thimblerig is 1 to break with many breakers, as you'd expect from a gear check, but Black Orchestra, one of the most commonly used decoders in Anarch decks, requires 3 to break a Thimblerig. This occasionally makes it possible to make Anarchs' centrals runs very taxing even when you don't have much money to rez ICE with: a Thimblerig plus two pieces of unrezzed ICE cost a total of 5 (2 in rez costs + 3 in install costs) and will tax an Anarch who relies on Black Orchestra 9 per run, which is an incredibly good ratio between the server setup cost and the break cost.
The main drawback of Thimblerig is that it has competition from gear checks that are designed to stop different sorts of shenanigans from those that Thimblerig prevents – in particular, a Thimblerig chain will force a Botulus to take a break every now and then in order to run its server repeatedly, but a Magnet will prevent the Botulus getting in at all. Magnet is also much more resistant to Aumakua. This means that if you're really trying to force a decoder install using cheap ICE, you probably want to be mixing Thimblerigs and Magnets. (At least they're only one influence each.) However, Thimblerig is still decently good at forcing the Runner's decoder to come out, which is something that will benefit almost any Corp deck due to the way in which it slows the Runner's deck down; you're making them find their decoder and pay to install it, and even if that's all the Thimblerig ever does, it has probably paid for itself. If it subsequently helps you put your ICE into improved positions, or maybe fixes damage done by Tāo, it has definitely more paid for itself. After all, it only cost 2 in the first place!