A more general solution: just use weights https://www.gravitricity.com

Be interesting to see how you could build this on a smaller scale for home use, using something very heavy but cheap like lead.

Lead is actually not that cheap. Lead ingots are $6/pound (at McMaster), while steel is $.50/pound (from Alibaba, but that's a minimum purchase of 1 ton). Scrap lead is around $.40/pound, versus steel at $.02/pound. The cheapest thing would probably be rock/concrete, or water.

A clip and pulley system which could hook onto a barbell or kettle bell would be great.

Hrmm, perhaps this isn't so feasible. Performing a clean & jerk of 95 pounds to a height of 7 feet would yield about 900 watt seconds of potential energy. So even if you had a very efficient laptop (15 watts) and perfect energy conversion, you'd have to perform a clean & jerk every 60 seconds to keep computing. Even if you were in good shape, you probably couldn't keep that up for more than a hour.

http://hyperphysics.phy-astr.gsu.edu/hbase/gpot.html

By cheap I meant not having to dig a tunnel, by using lead, which I think is the cheapest dense metal?, you avoid having to have a very long tether, say compared to water/concrete.

On a very small scale: http://gravitylight.org

Then how about sewage pumps For a combined energy storage and water distillation facility?