Cheap? Go to Farm Store or Nursery/Lawn/Home store and buy 2, 50lb bags of quality seed. Your choice. Rye, Taller fescues, Bluegrass, etc. What is best for you and for your climate. Seed investment <$200.
Put down 100 lbs of good fertilizer. Even pasture fertilizer is better than nothing. <$75.
Spread with $40 whirl type, walk behind fertilizer spreader. You'll either have that tool or will be needing one anyhow. Run a water filled roller over it, if you wish, to force ground contact of seed. Not necessary, but it helps.
Cover with 10-14 bales of good clean straw. $4 each or <$100.
You will have a spectacular lawn in no time. If you live north of the Mason Dixon, your time has just about run out for this year.
Your lawn, in a year, with proper care, will be indistinquishable from either hydro seeding or even sod. Remember, sod is merely grass grown from seed as well.
Less than $500 for the entire deal. YMMV. Done this upteen times with great success.