NMAC BTWN A DEP B737 AND A XING; APCHING CHEROKEE AT 500 FT AGL 1 NM SE OF HNL.

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AT 400 FT I CALLED FOR HDG MODE (TKOF). JUST THEN; TCASII ANNOUNCED A TA +1. LOOKED AT DISPLAY TO DETERMINE LOCATION. TARGET WAS SO CLOSE; TARGET WAS 'ON US.' SCANNED FROM 12 O'CLOCK POS TO 9 O'CLOCK POS. PIPER WAS AT 9 O'CLOCK POS AND CLOSE ENOUGH TO SEE COLOR (WHITE/BLUE STRIPE AND NUMBERS). I BANKED HARD R; OVER 40 DEGS; TO CLR AREA. ON TWR FREQ; I ASKED WHO PUT THE PIPER IN OUR TKOF PATH! ON DEP CTL WE DISCUSSED IT. DEP CTL STATED HE SAW IT ON RADAR. THE CTLRS WANTED TO KNOW IF I WAS GOING TO FILE 'PAPERS.' THEY ASKED 3 TIMES. THEN HE (DEP CTL) GAVE ME HIS NAME AND PHONE NUMBER TO DISCUSS IT WHEN WE RETURNED FROM OUR TRIP. SUPPLEMENTAL INFO FROM ACN 624539: WE DEPARTED RWY 8R AND DURING STANDARD TURN TO 140 DEG HDG; WE GOT A TCASII TA 'TFC; TFC.' I COULD NOT SEE THE TFC; BUT THE CAPT DID. ESTIMATE 45 DEG (R) BANK TO EVADE. WE GOT A GPWS 'BANK ANGLE' TWICE. I FINALLY GOT A VISUAL OF THE TFC. IT APPEARED TO BE A PIPER CHEROKEE. HE ALSO APPEARED TO BE TAKING EVASIVE ACTION. I NORMALLY AM VERY AWARE OF ANY LNDG TFC BUT WITH MY LIMITED EXPERIENCE IN THE ACFT WE TRUSTED THE TWR'S CLRNC WHILE I COMPLETED NORMAL CHKLISTS.