I am a very small (4'8", 85lbs) side-sleeping backpacker looking for a cheap sleeping pad. Because I am so small I need something lightweight, but it can also be very short in length (maybe a kids pad?). Because I am a side sleeper I need something thick. Thus, I am looking for a pad that is >3 inches thick, <1 lb in weight, but even as low as 50 inches length. I have only found pads with these specifications that are either very expensive (e.g Nemo Tensor) or have bad reviews (people say the Big Agnes pads leak). Any tips?
I am a big fan of close cell foam pads and I am a side sleeper myself. They can be easily trimmed and shaped to suit, theey cannot leak, nd they are relatively inexpensive. They do not have a lot of R value, so if that is significant, they must be supplemented.
I slep very comfortably, night after night....
I have two Big Agnes Air Core Ultra (insulated and not insulated) and neither leak. I've had one for over a year and the other for more than a few years. I recently switched to a Thermarest Neoair xlite to shave some weight.
Depends how warm you need it.
Getting a a pad that is >3in think that is cheap and light (< 1lb) and warm and reliable is unlikely unless you get lucky and find one used.
A good all around choice is the Therarest Xlite. The Small is 47inches long. A torso pad for many but may work for you. R 4.2 (good for deep 3 season in most places) 2.5 inches thick weigh 2 8oz. There is a Woman's version for a bit more that is warmer (R5.4), 5ft 6 long and weighs 12oz.
Not the most cushy pad and some complain they are noisy. I use a regular size and I'm a side sleeper, weight around 190-200lb and I think its fine. It doesn't bottom out when I'm sleeping even with the inflation slightly relaxed. It should provide plenty of support for a lighter person. The noise thing has never bothered me...actually can't say I have ever even noticed until people claimed it was an issue. Those sizes are out of stock most places currently unfortunately but the small usually costs $150.
I have not use this personally but other good comfortable light warm thick pads are the Nemo Tensor and the Sea to Summit Ether lite which come in various sizes. And yes, as you note, these are all on the more expensive side.
If you don't care about warmth (say you only need it for a summer non-alpine trip) then there are plenty of "cheap" inflatable pads that have an R value of maybe 1 that can work for summer use but are fairly light weight as a result. These can be found for less than $100, sometimes less than $50. The Klymit Static V2 is a good example. 2.5 inches ~1lb 3oz
I have not used one but some people have good luck with them.
Another apparently popular pad that Dixie of Homemade Wanderlust tested and liked https://www.youtube.com/watch?v=c1lzPuyX7bM
is the Sleepingo which weighs around 15 oz, is 2inches thick and generally sells for a around $40
Both of these claim a width of 23 inches that is slightly wider than the typical 20 inches of many light weight pads. They are also both longer than you need. It may be possible to shorten them if you are handy. I have seen this done with the Thermarest X-lite (...you need an iron) but I don't know if it works with these.
Beware claimed R values. Klymit in particular has estimated (over stated?) their R values in the past to the point that their "insulated" version may not be worth the additional weight according to some. They have been slow update their stated R values to conform to the current standard. Otherwise they seem to be fairly reliable. I have no information on the accuracy of Sleepingo's R2.3 claim but I suspect it is optimistic and is not measured with the new standard.
Any inflatable is vulnerable to puncture so test the pad before you go... (sleep on it) and make sure you carry a repair kit and know how to fix a leak.
Carrying an 1/8 closed foam under pad can be a good way to protect an inflatable, provide an emergency backup, give you a day/sit pad and slightly increase the R value .
As @hikermor suggests since you are light you might find a closed cell foam pad like the Thermarest Z-lite Sol or Nemo Switchback to work fine even though you are a side sleeper. They are lightweight surprisingly warm and actually moderately comfortable. They are bulky but can be easily cut down to match your height. They are also pretty much indestructible in an backpacking context.
Use an inflatable pillow like the Sea to Summit Aeros or similar.
What type hiking do you do/plan to do? 4 season (R5 pad) , 3 season (R3 Pad or higher) or just summer. That'll drive the options you have. One way to find new gear that is decent and not over used is to wait for an REI garage sale near you. You can find reasonable used gear on geartrade.com and if your just summer camping, DIY could be a great project where you shop locally, and adjust the materials to fit. I am a side sleeper as well and can't sleep on a foam pad as I find them uncomfortable. If you want to test that, most REI's have the closed cell foam pads for sale and you could test one in the store to see if it works for you. Good luck.