What is the common name or design pattern for a UI element which is like a combobox but has the added facility of typing to refine the options returned in the drop down.
If the field when entered displays a drop-down of all eligible values, and when the user starts typing into the field the options in the drop-down is reduced by hiding non-matching values, then this is a type-ahead filter.
A type-ahead field which does not display the drop-down when first entered, instead waiting for the user to type some characters before then providing a drop-down of suggestions is known as an auto-suggest field.
Some auto-suggest implementations will do a fuzzy match (see how my typing of "my ach" resulted in "my school" as a suggestion). This typically doesn't occur with type-ahead filters (usually because type-ahead filters are used for fields with a controlled vocabulary, not free-form text, and thus the range of options are defined and limited).
If the field when entered displays a drop-down of all eligible values, and when the user starts typing into the field the first matching value in the drop-down is selected (without hiding non-matching values), then this is a combo box.
Note that since non-matching values are not hidden the operator can use the arrow keys to move the selected option up or down. I have found this handy in that I might want to select the last of the options that start with the letter "S" and so I type an "T" and then use the up-arrow cursor.