Yes one of the most basic of mods that any car enthusiast will do, the trusty cold air intake. Find yourself a pod filter and some lengths of pipe and mount your intake somewhere that gets cool air. Just be sure to avoid a location that may result in your engine inhaling water, and avoid hot areas like the exhaust system. While there are manufactured kits available, generally they are quite pricey, and a diy setup can be put together for around $50 and is a great first step for someone who wants to start making their own car mods.
The second on the list is again a very popular choice and perhaps no suprise, the exhaust headers. It will free up that exhaust flow, improve engine performance, and you'll maybe even gain a bit of a tougher exhaust note. This is a great one to try and find used on ebay or your favourite spare parts marketplace. A used set of headers will come in at somewhere between about $100 to $300 dollars. Installation is definietly within the capability of even the most amatuer home diyer with just simple tools such as a basic socket set.
Third on the list is the next part down the chain attaches to the previously mentioned headers, the EXHAUST. A free flowing or straight through rear muffler will again improve that exhaust flow, will definitely add some tone and noise, and may improve power. Search around and you can often find a used one for sale for a fair price, just be sure you get the right exhaust for your year of car as they did change over time. Generally tubing around 2 and a quarter to 2 and a half inches in diameter is a suitable upgrade for an MX-5 miata so keep that in mind when your on the hunt.
Number four is a free mod for those with the original NA model MX-5 Miata that run the mechanical crank angle sensor, or cas, at the back of the head. The 14-degree timing mod. All you need is a timing light which can be purchased or sometimes hired from your auto parts store, or perhaps just borrow one from your mates, and a spanner or wrench to loosen and adjust your CAS. Advancing your base timing from 10 to 14 degrees before top dead centre is a free trick to unlock some power at the top of your rev range, note that higher octane fuel is advised to avoid the changes of engine damage with more advanced timing. Theres a great guide on the miata.net website that details all about this mod, https://www.miata.net/garage/ignition.html#timing
Number 5 is an exclusive for the 1.6 litre engined vehicles with the flapper door style Air Flow Meter. This trick involves swapping in an air flow meter from a non-turbo 2nd-gen RX7, which can be had from your local used parts junkyard. This will unlock an additional maybe 5 or so hp, but this benefit will really only be present above 6,000 rpm so don't forget to rev that 1.6 all the way out to redline! A great web link on this topic can be found at the solomiata website: http://www.solomiata.com/airflowmtr.html
Sixth on the list is another freebie. Don't let unneccesary accessories sap away at your already measly power level. Why not consider an air-con or power steering delete. Both of these systems require belts and pulleys running off the crank of the engine, and as such are taking away valuable horesepower that you desperately need under your right foot. Simply removing the drive belts to these systems is one quick, easy and free power adder.
This one might be a suprise, but for number seven, believe it or not, spending maybe $100 bux on a fresh set of ignition leads and spark plugs can do wonders. Resolving a weak ignition system, even if you didn't notice a power defecit, will make the whole engine feel brand new. It's an easy DIY job that takes maybe 20 minutes with parts easily available at almost any auto parts store. I've personally driven numerous MX-5's in my time that had noticably weak power under load that were fixed with fresh plugs and leads.
From here on, we're starting to get to some of the more serious stuff. For number 8, it's a head skim. This DOES require removing the head from the engine, and taking it to a machinist. But if you are handy and remove your head yourself, you'll only be out of pocket for the cost of the skimming which would be in the viciinty of a couple of hundred dollars. The benefit is an increase in compression ratio, and can gain some 10 to 20 horesepower, which is really pretty significant. A modification like this does need to be complemented with other engine power adders to make it worth the while, so best not to start with this one and get some of the basics earlier in this list out of the way first.
Ok I really don't want to get into the 1.6 vs 1.8 argument, but number 9 is the option of swapping in a wrecker 1.8 engine in to replace your worn out and tired old 1.6. With the availabliity of conversion kits from people like Flyin Miata, for about $250 you get everything you need to be able to swap a 1.8 into the 1.6 vehicle and running without too much fuss. This is particularly handy for those who have an engine thats on it's last legs, and want the extra torque of the slightly larger engine. And, given that the 1.6 chassis and driveline is ever so slightly lighter and quicker geared, it'll give the car some noticable pep.
Last but not least, forced induction has to be on the list! Not exactly cheap, let's be frank, but you CAN do boost on a budget if you really want to. Find a used supercharger or turbocharger kit, something simple and basic and run low boost, and you wont need to upgrade too many other OEM components. This is one of those situations where, if you bide your time and keep an eye on your online marketplace of choice be it ebay, facebook or the forums, you'll find someone offloading a kit for maybe 500 to a 1000 dollars, you can very quickly add quite a lot of power.