If at first you don't succeed...

...then try again. Then try again more 125 times. Then quit.

PROCEDURE get_id
    ( p_id_out         OUT NUMBER
    , p_name_in        IN VARCHAR2
    , p_create_user_in IN VARCHAR2 )
IS
    v_new_id      NUMBER := 0;
    v_max_tries   PLS_INTEGER := 127;
    v_default_id  NUMBER := 0;
BEGIN
    v_new_id := lookup_id(p_name_in); -- will be 0 if not found

    WHILE v_new_id = 0 AND v_max_tries > 0
    LOOP
        BEGIN
            INSERT INTO entry
            ( entry_id
            , entry_name
            , create_date
            , create_user
            , create_app
            , mod_date
            , mod_user
            , mod_app)
            VALUES
            ( entry_seq.NEXTVAL
            , p_name_in
            , SYSDATE
            , p_create_user_in
            , 'get_id'
            , SYSDATE
            , p_create_user_in
            , 'get_id' )
            RETURNING entry_id INTO v_new_id;

        EXCEPTION
            WHEN OTHERS THEN NULL;
        END;
    
        v_max_tries := v_max_tries - 1;
    END LOOP;

    p_id_out := v_new_id;
END get_id;

Thanks BB for sending this.