I have a Celestron Firstscope telescope and like it overall for my location and the amount of observing I do. I am disappointed in my view of the planets with the scope. What would be a good eye piece to purchase for this telescope to improve planetary observing. In keeping with my limited budget, it should not cost more than the telescope.
I'm not sure what eyepieces come with the Celestron FirstScope. I have the very similar Orion FunScope, which came with 2 eyepieces: 20mm and 10mm focal length, yielding 15x and 30x respectively. I have found that the 10mm is around the upper limit of what this telescope's optics will support, and would not recommend any eyepiece giving higher magnification. In fact, most of the time, I use only the 20mm because the views are better.
Small telescopes like these are generally not capable of very high magnifications. To get a satisfactory view of the planets you need to spend more than $50. This telescope is fine for what it does, wide field views of the Moon, planets, and large deep sky objects, but simply is not capable of the magnifications (150x to 250x) needed for good planetary views. For that you need at least a 4-inch refractor or a 6-inch reflector.
I tried a 5mm Hyperion from Baader Planetarium (great eyepiece) on the FirstScope of a colleague but the thing was too heavy and it became difficult to keep the telescope stable. So since you're dealing with an extremely light telescope you should also take weight and balance issues into account.
There is an additional lens, called a Barlow lens, that can be added in front of an eyepiece to increase the magnification by 2x or 3x. It's basically a negative power lens that reduces the convergence of the rays from the primary, thus simulating a longer focal length. They're cheap, so you might try one with your Firstscope just to see if it helps. Of course, any imperfections in the primary mirror and wobbles in the mount will also be magnified.
Here in the UK the SkyWatcher "Ultrawide" series fits your budget and is a decent improvement over the stock eyepieces included with budget telescopes. The branding is rather misleading, as "Ultrawide" conventionally refers to eyepieces like the TeleVue Nagler series that have greater than 80 degrees apparent field of view (AFOV); the SkyWatcher ones are significantly below this. However, despite this, they are reasonable quality, have longer eye relief which makes them comfortable to use, and perform acceptably on-axis at f/4 (off-axis performance suffers a bit, but f/4 is very demanding and only very expensive eyepieces retain good performance to the edge of the field in very 'fast' telescopes).
They are available here used for well under £20, which suggests that they would fit your budget if they available in the US.
Another good option are clones of the Burgess/TMB planetary design, which are again long eye-relief designs optimised for planetary performance. They're a little more expensive, but should be at the upper end of your budget - here they're about £25-30 used. The Burgess/TMB-branded versions have better cosmetic finish and are more expensive, but the actual optics appear very similar.
Another issue that hasn't been mentioned yet is collimation - it's very important for a fast Newtonian, and if the telescope is out of collimation it won't matter how much you spend on an eyepiece, the views will never be good. I found my son's Firstscope rather hard to get right as adjustment of the secondary mirror wasn't easy, but got there eventually.
There are lots of guides to the process online, I used
It's on the far end of your price range, but the "Super Abbe" line from University Optics has a good reputation. You don't say what size your current eyepiece is, but their 7.7mm would probably be a good bet. You don't want to go too high on magnification, as you'll have trouble keeping things in view. When looking at Saturn with my 6" Newtonian and a 5mm eyepiece, it drifts out of my field of view within about 30 seconds.
Zhumell from telescopes.com makes a great kit that you will not out-grow as you can use it on up-grades. But for just one lens that's higher quality but not to expensive, I love the Zhumell 12.5mm. Just got through looking at a double star cluster in front (South) of Cassiopeia. Just brilliant on the firstscope.
You're not going to get much more than that out of this instrument. It's cheap, it's a very fast f/4, and to top it off - it's spherical! At f/4 even a well-corrected parabola would be very challenging for any eyepiece except cream of the crop $400 glass. A sphere mirror only adds its own issues on top of that.
Short instruments look easy, but are tough on the eyepiece. People should stick with f/8 or longer for their first scopes. That's not difficult for any eyepiece, and it doesn't matter if it's a paraboloid, or a "mere" sphere.
Maybe try a decent Plossl eyepiece, there's hope they are better than the originals. But don't expect miracles.