Solution 1:

I would call it a selector, since it is used to select an option in the menu.

Google defines selector as:

a person or thing that selects something, in particular.

Also, "highlight character" limits you to a single character. What if you wanted to change it from ">" to "->"? Then "highlight character" wouldn't be accurate since it's more than one character.

Solution 2:

"Highlight character" doesn't make much sense to me.

Presumably, if you press Enter, that is the menu item that responds. By definition, that's a cursor.

In computing, a cursor is an indicator used to show the position on a computer monitor or other display device that will respond to input from a text input or pointing device.

Solution 3:

The name is the name of the particular character used (if a character is used). There is no general term that I am aware of. In some lists on a computer form, one or more options may be selected by default (e.g. by a background colour, by bolding, or by an adjacent symbol such as an asterix to indicate this to the user). Bolding or colouring are not characters but do service to 'highlight' that the item is 'selected'. More usually though the use of a background colour around text or symbols on a computer screen is called "highlighting".

Solution 4:

In this context, I would call it an arrow, pointer or highlight symbol.

I should stress, however, that these are my own personal usages.

I suggest you try soliciting an industry-specific answer from one of the StackExchange computer-related Q&A sites, such as the User Experience site.