IMO - Either the Hinsdale or Downers Grove stop on the BNSF

The Burlington Northern does stop at Hinsdale, but then there's not an easy way to get from there to Oakbrook by public transportation.  Hinsdale is just south of Oakbrook.  There may be a stop in Elmhurst with a different train line, but I don't know that area as well.

Hope that helped a little.

Downers Grove maybe but then it is still a cab ride.  I'd check out buses from the city if you don't want ot rent a car.

The Union Pacific West UP-W Line to the North that stops in
Elmhurst Station:
   128 W. 1st St. (at York Rd.)
   Elmhurst
Or
Villa Park Station:
   349 N Ardmore Ave.
   Villa Park,IL

And the Burlington BNSF Line to the South with stops in:
Highlands Station:
   1/4 mile North of the Intersection of County Line Rd. & 47th St. Hinsdale
Or
Hinsdale Station:
   21 E Hinsdale Ave. Hinsdale
Or
West Hinsdale Station:
   Hinsdale Ave. & Stough St. Hinsdale
Or
Clarendon Hills Station:
   1 South Prospect Ave  Clarendon Hills


http://metrarail.com/content/metra/en/home/maps_schedules/metra_system_map/up-w/map.html

http://metrarail.com/content/metra/en/home/maps_schedules/metra_system_map/bnsf/map.html

I have a place in downtown and in burbs.  Are you closer to Union Station or Ogilvie in Chicago and when and where are you going in Oak Brook.  All things being equal, West Hinsdale has an express train during rush hour which takes just over 20 minutes from Union Station.  And by cab, going up north on 83 to the Oak Brook Mall area is faster than going south down York to 22nd Street area from Elmhurst station.