Creating simple maps to view on Mobile devices?

You're right to look into Mapbox, despite not having a Computer Science background. Mapbox has a utility called TileMill which is essentially a web map development studio that has a familiar GIS-like interface. It is based in Javascript, but is designed to make development easy for non-developers. It gives you all of the code as well, though you may or may not actively use it.

https://www.mapbox.com/tilemill/

You have a few alternatives for simple, free maps:

• Google Maps Engine, upload your data, style it, servie it up as a simple map with an iframe to embed it in a website, or tiny code snippets. Very easy, point-and-click interface, no javascript
• ArcGIS Online, similar to Google Maps Engine, just limited like you say
• MapBox, you can upload a simple dataset that was styled in TileMill, with popups and everything nice. Then you can link and embed that map. It's really easy to get nice maps. Here's an example: http://a.tiles.mapbox.com/v3/alexgleith.map-m2406w0i/page.html

For the most flexibility, I like Leaflet. You need data stored somewhere still, though. So, for static maps, I'd recommend MapBox. You can also try Google Fusion Tables, CartoDB, or any number of others. MapBox is the fastest to make something pretty, though.