In my Hello posting I used a couple different platforms to make my map. I made a mental note to go back over these, the benefits and drawbacks, of each for the RRSI participants and followers. I don't have to from scratch. Someone else did.
Elisa Beer (@FordhamGSDH) wrote a blog posting entitled "Mapping Religious Concern in the Later Middle Ages: Software Ups and Downs for DH Visualizations" in which she has a similar mapping scenario of a several hundred manuscript tradition. She weighs in on Google Map Engine Lite, CartoDB and Microsoft MapPoint. I have not used the latter, but I tend to agree with her about the relative benefits of Google Maps and CartoDB. Google Map requires you to have a Google account, and it does not handle multiple points in one location well (pins do not aggregate into larger pins), creating the illusion of a scarcity of points in my second map. It also limits you to three layers, with 100 points per layer, for a maximum of 300 points. The legend section is not customizable. For some quick and dirty humanities mapping this is sufficient. The data for my Visualizing Medieval Places project has reached 5000 points, but most of the quick maps I make have only several hundred points. Not exactly big data. I like the term "Pareto GIS" that Harris, Rouse and Bergeron use in their chapter in The Spatial Humanities: GIS and the Future of Humanities Scholarship.
Google Maps also allows you to sharing a map as you would a Google document with others with an account. When you upload your map, your data is not public. A student of mine @Randa_DH used it to make a map of the places (imagined, visited, mentioned) from Barghouti's Arabic novel Rai'tu Ramallah. NB: You need a google account to view this.
CartoDB is available in an academic edition, so if you are not publishing maps and getting lots of hits or especially if you are doing your own georeferencing, I think CartoDB is the way to go. It allows for a number of filters and SQL queries. When you make maps using the free academic version, your data is public.
There are others. Y! has a tool Map Maker that allows a small table of points in Excel to be loaded into Yahoo Maps.
Map Builder allows for quickie mashups, but I have never tried it.
If you are actually moving around in real space and you want to create a map with your smartphone in hand, a good app to download is Click2Map. Another student of mine @AliceKezhaya mapped the various languages of signage in our multlingual neighborhood in Beirut. Her blog posting about that can be found here.
All of these resources allow you to plot onto a regular google-style Mercator map without learning ArcGIS or QGIS. If you learn those with a certain degree of fluency and you want to plot onto historical basemaps, the place you must learn about is Map Warper from the New York Public Library.