The MacBook Air Original (Early 2008) A1237 uses 1.8" IDE drives with ZIF connector as a storage device.
The HDD are slow, not reliable and start getting expensive as working ones go rarer.
There are some 1.8" IDE SSD available on Aliexpress, but they are still expensive for what they are (more than $50 for 64GB).
One solution could be to use an mSATA to IDE ZIF adapter board, but unfortunately this does not work. They use JMicron JM20330 controller, and it does not seem to be recognized at all on the MacBook Air Original.
Another solution is to use an SD card inside an SD to CF adapter, and plug that adapter into a CF to 1.8" IDE ZIF adapter and this actually works. Both adapter should cost less than $12 in total, and then you can get a good SDXC 64GB card for less than $18, so you get 64GB of Flash storage for less than $30.
Note that the MacBook Air Late 2008/Mid 2009 A1304 uses SATA drives so this method will not work.