What's the best (least expensive) way to stop water from eroding?

you might install an under ground pipe from the gutters to a side yard, there are French drains as well. You can dig a dry creek bed to channel the water around the house.

A French Drain that channels water out to the street storm drains, then you can also add an outside sump pump, catchment basins and pop ups. I suggest you watch a bunch of Apple Drain videos to see what will work for you.... https://www.youtube.com/watch?v=InqitNqdx5g

pop ups and catchments.... https://www.youtube.com/watch?v=y9_z9Zlle8o&t=199s

sump pump and French Drain... https://www.youtube.com/watch?v=ywgDOSHvsIg

If you want to “try before you buy”, just dig a 6-12” ditch to divert the rain around the house. See how the water settles on the other side, if enough rain was diverted to make a difference, if the ditch was a safety hazard, and other issues that came up. After analyzing this information, you can decide on the next step. Best wishes ☺️

French Drains will do what you need. Dig a trench along the outside of your footing. ...

Lay the pipe on the virgin soil. ...

Cover the pipe with at least 12 inches of washed gravel.

Lay filter fabric over the gravel to prevent any soil from clogging the pipe.

Back-fill the foundation with top soil back to its original grade height.

To stop the water pooling by the house bring in soil to build up the grade. This will make the run off go away from the house