First, a quick primer on the train options in Chicago. There are two train systems, the Chicago Transit Authority (CTA) and Metra. The CTA is mainly designed to shuttle people around within the city of Chicago; Metra is mainly designed to shuttle people from the surrounding towns into the downtown core and back out again (i.e., "commuter rail".) The CTA has a terminal right inside O'Hare airport; Metra has a station somewhat outside of the airport, with a shuttle bus that runs to & from the station.
With that in mind, here's my take on the two AirBnB locations you mentioned:
Bensenville is a suburb of Chicago, just south of the airport. Public transit there would be a real hassle. You'd have to take the shuttle bus to the Metra "O'Hare Transfer" station, take the train towards the city a few stops, get off, transfer to a different line going back out towards a different suburb, and then get off at the Bensenville stop. From there, you'd have to walk to wherever the AirBnB location is; on the map, it looks like it probably wouldn't be more than a kilometer, but that's a long distance when you're toting bags. However, getting from O'Hare to Bensenville with a cab from the airport would be relatively easy; it wouldn't even require getting on the freeways, which would be a real plus if you're arriving around rush hour. Cab fares are available here; the distance from O'Hare to Bensenville is about 8–10 miles, so the total fare would work out to be in the $25–30 range (before tip, which would be expected; this also assumes no time stuck in traffic, for which you'll also be charged.) Note that since Bensenville is outside the Chicago city limits, there's a 50% surcharge tacked onto the fare.
Irving Park is a neighborhood of Chicago on the north side. The CTA Blue Line has one terminal in the O'Hare Airport, and would take you directly from there to the Irving Park neighborhood. However, from there you would still need to walk a substantial distance to get to Elston Avenue. For a cab ride, the distance appears to be about 12 miles, which would yield a fare of about $25 (again, before tip.)
Given that your ride from the airport will be at 10 AM (at the end of the morning rush hour, but still with substantial traffic) and that your ride back to the airport will need to be at around 6 PM (definitely during the evening rush hour), I would go with the Bensenville location. Chicago traffic can get pretty bad, and I would place a premium on being able to get to and from the airport quickly.
Also: hailing a cab at the curb may not be your best option in this case; you might want to look into a car service (aka "limousine service") and pre-arranging a ride to & from the airport. For the amount of people and luggage that you have, it would definitely save you hassle and might even save you money. However, I have zero personal experience with doing this, so I'll let someone else speak to the feasibility of this option.