You might have better luck at IAD main line United customer service, rather than the G gates customer service. Also next time when this happens, go to the baggage service office at LGA or wherever you are arriving. Baggage service offices have the facilities to issue taxi and probably hotel vouchers, they can rebook flights, and do any number of other things. They don't have ticket stock though.. Sometimes at smaller stations they as a matter of procedure send pax to baggage service to get these vouchers printed out. I might have asked at LGA baggage service either for a cab voucher to HPN, a car service, or if it was really that bad, a hotel at LGA and then ground transportation in the morning. Or again perhaps the hotel at IAD and then flight out the next day to HPN.

I am confused as to why people put the burden on the pax not to book these tight connections. The airlines sell these connections. Sure we all know they are tight, but I don't understand the idea that if the airlines sells you that connection that they should not provide the same ameneties as say on a 3 hour misconnect. Now mind you if I really had to be somewhere that night or the next morning, this might not be the best connection to book, but the airlines all sell notoriously tight connections and technically they are legal connecting times. How is a once a year flyer to know about connection logistics, times, etc. If the airlines sell these connections then they should have the procedures in place to provide ameneties in the likely event that these connections will often be too tight.

You may get a $100 voucher if you write to customer service. That is pretty much standard compensation for almost anything. But I would agree, next time try like heck to deal with it at the airport so you don't have to be writing letters and hoping on a prayer they send you something.