Current and estimated arrival times
A couple of things I noticed:


No Identification Division. "The IDENTIFICATION DIVISION has no effect on the execution of the program but is, nevertheless, required as a means of identifying the program to the computer" - Stern &amp; Stern
There is no real reason for the first REDEFINES. Also, you seem to be using it wrong. A redefines clause is simply another way to reference the same working storage location. If you want to store different data, that will require different variables. While you can keep redefining the same variable and using it in different way, it can get messy and storing things gets harder.
While you can have level 3 variables, I have found it to be best practice to go up in increments of 5
A lot of the time, if you are working in a real COBOL environment, you will need an environment division. Clearly here you don't, but keep that in mind
You should start all variable names in the WORKING-STORAGE SECTION with WS- because if this program need to interact with any other programs, you will have a LINKAGE-SECTION with its own set of variables.
Only use periods to end paragraphs. Technically, you can use a period to end anything. This leads to messy code. Also, a period closes ALL open statements. If you have a triple nested IF and you end with one period, it will close all 3. When you get to that point, use the proper scoop terminators such as END-IF and END-PERFORM
Spacing is important as well.


The logic problems have already been address in another answer, so I will not address that here. I have included the code below with my suggestions added to it:

        INDENTIFICATION DIVISION
        PROGRAM-ID. THE-TIMER.
        AUTHOR. name.
        DATE-WRITTEN. date.
        DATE-COMPILED. date.
        *****************************************************************
        *header comment about the program change logs ect               *
        *****************************************************************

        DATA DIVISION.
        WORKING-STORAGE SECTION.

        01 TIME-STORAGE.
            05 WS-CURRENT-TIME        PIC 9(8).
               10  WS-CURRENTHOUR     PIC 99.
               10  WS-CURRENTMINUTE   PIC 99.
               10  WS-FILLER          PIC 9(4).

            05 WS-ARIVAL-TIME         PIC 9(8).
               10  WS-ARRIVALHOUR     PIC 99.
               10  WS-ARRIVALMINUTE   PIC 99.
               10  WS-FILLER          PIC 9(4).

            05 WS-ETA                 PIC 9(8).
               10 WS-ETAHOUR          PIC 99.
               10 WS-ETAMINUTE        PIC 99.
               10 WS-FILLER           PIC 9(4).


        PROCEDURE DIVISION.

            BEGIN-BUG-NORMAN.

            DISPLAY "when will you be ariving?"
            DISPLAY "HH [enter]"
            ACCEPT WS-ARRIVALHOUR
            DISPLAY "MM [enter]"
            ACCEPT WS-ARRIVALMINUTE

            ACCEPT WS-CURRENT-TIME FROM TIME
            SUBTRACT WS-ARIVAL-TIME 
                FROM WS-CURRENT-TIME
                GIVING WS-ETA

            DISPLAY "RESULTING ETA: "
            DISPLAY ETA-NUMERIC
            DISPLAY "HOURS: "
            DISPLAY CURRENTHOUR
            DISPLAY "MINUTES: "
            DISPLAY CURRENTMINUTE

            IF WS-ETAHOUR &gt; 5
               DISPLAY " It could be a while"
            END-IF

            STOP RUN.