Is there a way to modify the standard pop up menu shown when someone right click in a windows text box?

Override CWnd::OnContextMenu by adding a WM_CONTEXTMENU handler to your CEdit derived class. From within this handler, you must load your own popup menu, call TrackPopupMenu and call the menu functions according to the returned ID.

Example:

void CMyEdit::OnContextMenu(CWnd* /*pWnd*/, CPoint point)
{
    CMenu menu;
    VERIFY(menu.LoadMenu(IDR_MYEDIT_POPUP_MENU));
    CMenu *pSub = menu.GetSubMenu(0);
    // When right clicking on a control, the focus is sometimes not moved to it
    //  before calling this function.
    if (GetFocus() != this)
        SetFocus();

    // if opened by keyboard (not mouse)
    if (point.x < 0 && point.y < 0)					
    {
        CRect rect;
        GetWindowRect(rect);
        point.x = rect.left;
        point.y = rect.bottom;
    }
    int nCommand = pSub->TrackPopupMenuEx(
        TPM_LEFTALIGN | TPM_RIGHTBUTTON | TPM_VERPOSANIMATION | TPM_RETURNCMD | TPM_NONOTIFY, 
        point.x, point.y, 
        AfxGetMainWnd(), NULL); 
    switch (nCommand)
    {
    case ID_EDIT_COPY : OnCopy(); break;
    case ID_EDIT_CUT : OnCut(); break;
    }
}

For this you have to intercept messages that sent to the edit box you need to
replace its window procedure with your own. This technique called
subclassing.
You can read about subclassing here : Windows subclassing and hooking with C++ classes[^]