Here is the cheapest way to be something like backpacker in Saudi Arabia.
I did it before couple of years :)
You should have membership of the "Saudi Arabian Youth Hostels Association"
The membership is for 200 Riyal=53 dollar for a whole year. (There is no other options only whole year).
You can have it if you are older than 15 years.
And you can stay in each city for 30 Riyal/day(less than 8 dollars) .
And there are houses in each large city in Saudi Arabia(21 house).
And you can have your kids with you also(male kids only) if they are older than 10. With the same fees.
Here is their website:
http://sayha.org.sa/en/
And they have a relation with hi hostel.
You can book for your stay from their website, but you still have to be for the membership fees after your arrival:
https://www.hihostels.com/destinations/sa/hostels
And they will give you a bed in males shared dorm room.
If you stay outside Riyadh you might be the only guest they have, because not a lot of peoples are using it :)
Another Option:
Also if you are not traveling alone, it might me cheaper to find cheap private room for rent in cheap "Furnished Apartments".
There is no Motels in Saudi Arabia.
There is Hotels or "Furnished Apartments".
"Furnished Apartments" is usually cheaper than the hotels you might find luxury Furnished Apartments for 800+ SAR or you might find cheap one for 40+ SAR.
As I said you might find Cheap Room in Furnished Apartments and they rent it for 40-50 Riyal But this is not even equal to 1 star hotel.
and you can't book for it online, only face to face most of times :)
In each city there is a neighborhood Full of cheap Furnished Apartments:
Riyadh You Can find some in "Batha neighborhood".
"Eastern Province" in Khobar there is "Al-Thuqbah neighborhood".
Jeddah you should go to South of Jeddah.
Madinah in the North of King Faisal Rd, near Almasani neighborhood.
Buraidah Next To Alkhubaib Street.