To customize web messenger user avatar, you need to override the default function (ChatUI_GetInstantAvatar).
Please add the following script into the bottom of messenger.aspx.
function ChatUI_GetInstantAvatar(user)
{
return "DownloadMyUserPhoto.ashx?UserId="+user.UserId;
}
In the above code, you need to create the DownloadMyUserPhoto.ashx, and redirect it to your own photo's url.