Members wouldn't be able to do that. The avatars are displayed in a set container that can only be changed through css adjustments. You would need to adjust the container, and the photo size, and would most likely need to use the "!important" tag since the base css will try to size the avatar to the default sizes. There are several avatar images defined. Be sure your working with the right one.

I think the easiest solution is to use the design settings plugin and have your members paste some CSS to change the avatar size. Everybody that visits a specific profile will see the avatar as the size that the person set in their CSS.

In the custom CSS box in that plugin, you'd paste the following if you're using the Simplicity theme:

div#avatar_console_image {

    height: 80px !important;

    width: 80px !important

}

Or this for the older theme:

div#avatar-console > div {

    background-size: 80px 80px !important;

}

Then the user can change the height and width px value to make the avatar larger or smaller on their profile. If you're using a different theme, it can probably be done, you'd just need to take a look at the avatar CSS.

EDIT: I haven't used that plugin, but I did test in on his demo site. Worked fine.