A simple but possibly insufficient solution is to use a formula. If we were to assume that you already had a list of unique names in the A column, you could fill another column with this:
=AVERAGEIFS(score,name,"="&$A1)
If we need to create the list of unique names however, we’re going to have to use a pivot table (or macros; or a really hacky formula solution).
Select the two columns, click Insert → Pivot Table… → Current Selection → OK. Drag Name over in the Row Fields list, and drag Score over in the Data Fields list. Double-click on Sum - Score in the Data Fields list and select Average from the Function dialog.
(If this works for you, please click on the ✓ to accept the answer.)