moving into a house with an elevated deck, with an unfinished walkout
basement underneath the deck. The decks sits up some 10-12 feet.
Underneath the deck right now is grass/dirt/mud. Pretty nasty. I
have long term visions of a finihsed walkout basement leading to a
nice patio.

However that is the future, and this is now..

is there a low cost, DIY way to clean up the area underneath the deck,
such as adding a gravel patio? If so, how would one accomplish this

We too had thoughts of such a wonderful lower level patio when we bought this
house more than a decade ago.

There was some large gravel under the deck and we added small gravel to fill in
and make the walking easier.

Since there was much to do to landscape the yard, under the deck became the
potting/nursery area. The deck provided shade during the hotter hours of the
day for both me and the young plants I was propapating. Since the deck faced
east, the plants got good early sun when placed near the outer edge of the
lower area but didn't get the harsh afternoon and west sun (Zone 7-8).

Be sure the slope is away from the house to prevent water issues.