Likely you have done is taking Oedo Line from Shinjuku to Daimon, then Hamamatsucho to Haneda Airport on Tokyo Monorail.
Likely scenario is that when you got off at Daimon station by some reason your Suica card did not register as you got off at Daimon station. Did you swipe your Suica card as you walked out of Daimon station? Not having record of getting off at Daimon station is likely reason your Suica card was not accepted at monorail Hamamatsucho station.
Also, traveling from Shinjuku to Haneda Airport more common routing is JR Yamanote Line from Shinjuku to Shinagawa, then Keikyu Line from Shinagawa to Haneda Airport.
On your routing using Suica was total fare of 699 yen, where JR Yamanote line and Keikyu Line routing using Suica is 601 yen.