As the remnants of Lilli came over our area I watched from the 25th floor as a circle of light surrounded by the rains passed by. In the area of this clear sky there were hundreds of birds. Small birds that are rarely, if ever, seen at this height. They were flying in circles, arcs and tangents, only within this non-rain area.
Was this the remains of the eye?
Why the birds? Were they pulled up? Do they have some inclination to fly into that circle of calm? Or were they surfing?
I think we need more info. What city are you in? If Lilie was no longer a hurricane, it would no longer have an eye. How large was this fair area? An eye normally is large enough so that you cannot see the surrounding rains. (In other words, there would not be a circle of light.) Further, in the eye itself, the air is sinking, so the birds would not be pulled up.
That said, the birds were probably flying there because it wasn’t raining there. After all, they are not bird-brained. 
I live just outside of Cincinnati, OH. The area of light sky was totally cloudless. Somewhat eliptical. It was perhaps 30 yards wide.
Distance can be deceiving from the 25th floor, though.
I rarely, if ever, see small birds from my office. Generally a crow, or perhaps a few martins gliding on wind currents. These were definitely small birds. Sparrows, wrens and the like. They circled the area of no rain in a counter clockwise direction. They were definitely staying within the walls of rain. There were many different circular arcs, up and down. But they didn’t cross each other. There were hundreds of birds in this area. And they flowed with the movement of the calm area from SW to NE.
I’m guessing that what you saw was a flock of blackbirds (either starlings or grackles, most likely) that just happened to be flying in a crack in the rain clouds. An area 30 yards wide wouldn’t be big enough to qualify as the “eye” of a hurricane, and anyway, as Lili came up the Mississippi Valley heading towards Iowa, she broke up into miscellaneous areas of scattered showers. Cincinnati would have had only sporadic rain.
And the beginning of October is prime “flocking” season for starlings and grackles. They’ll spend the winter in these flocks, roosting in trees in parking lots, getting droppings all over your car. 
What you saw was just a flock of them temporarily caught between two batches of rain clouds.
I’d buy your theory Duck Duck Goose, except for two things:
I grew up on a farm and know what both starlings and grackles look like. In fact we have starlings year round and my grandfather used to pay me to kill them when I was a kid. They messed with crops.
These were definitely small birds. My office overlooks about a 3 square mile area of downtown. All small buildings. And I almost never see small birds. In fact the reason I so noted this event was that I was seeing small birds. Besides the fact that they were definitely flying in this particular circular pattern.
As to the rain that came through here that day, it was hardly scattered showers. Perhaps 1 to 2 inches in a very short time. It may not have been the remnants of Lilli. Perhaps the cold front that followed through. . .but the wind was definitely SW to NE. . .our usual pattern. Iowa is north of us. . .Generally an approaching cold front will pull southern air in front of it up the Ohio Valley.
As I indicated, I grew up in the country. I currently live in an area of town which is declared a bird sanctuary. And in my 50 some years I’ve not seen our feathered friends act like this.
Well, okay, they were a flock of small migratory birds, then, thrushes or warblers or something, caught in between two different batches of rain clouds.
<< shrug >>
Lili merged into a trough that was crossing the eastern half of the country. In fact, it quickly lost tropical status and was indistinguishable from the baroclinic low pressure area heralding the cold front. During migration season, many birds take advantage of these fronts as an aide to migration. In fact, during the spring and fall migration periods, birders look eagerly to a front’s passage for new migrants. Surprisingly, birds that migrate from north to south will take advantage of a front that moves south to north, and vice versa. I really don’t know how they do that, but they do. Since some warblers may be returning from southern lands, they could have taken advantage of the front.
BTW, when you are in a hurricane (which you, of course, were not), the winds will suddenly diminish and the skies will clear. This will last for a while, before the opposite occurs, the length of time depending on the width of the eye, measured in miles not yards.
I should add that you may have had some heavy rains even tho it was no longer a tropical system since it still had tropical moisture. Unfortunately, from what I heard, further east from you, where they desparately need rain, they did not get much. I was really hoping upper NY state, etc. would get a lot of rain since they are in an extreme drought.
Thanks barbitu8. I had no illusion that what I saw was an eye. I didn’t know if such atmospheric events could deteriorate and continue moving. It was a very distinct circle of apparent calm in the middle of a deluge. But from the way the birds were flying it was obviously not calm.
Perhaps just a freak of the two storms coming together that allowed the migrating birds to play centrifigal force games in the sky.