I have a less than one acre site to survey with tape and/or theodolite. I want draw a plan with heights as contours or spot heights. What cheapo and easy software is recommended?

Well open source software is always a good start, for cost. But it really depends on what you want to do; position boundaries, exact heights, maps trees etc.
As for easy, again depends on what to do, the more complex and powerful the steeper learning curve. Most cave software is not brilliant for surface, a GIS is what you probably want., Curtain what I used for plotting trees in an orchard, that and a distox2.

If you just want to draw a 2D site plan with spot heights, I would thoroughly recommend Draftsight, which can be downloaded for free. http://www.draftsight2016.com/

Unlike most free software, this is a finished product (made by Dassault Sytemes who mostly do top end 3D modelling/ simulation software).

I have used it for work for several years and while it doesn't have all the gimmicks you get in AutoCAD, it will certainly do everything you need. If you are keen, you can plot your points in 3D space and automatically generate sections from your plan view.

If you want to plot contours, you'll have your work cut out surveying with a tape and theodolite though!

You could download the free lidar data if its in open land and it is the surface height you are interested in ? Might save a bit of work...I think you can get 0.15m grid interval

Check Topographer or Topographer Pro for Android.