If we had a natal chart for the Olympics, then we might be able to speculate.
Two possible 'birth' dates and locations are:
8th August 2008, Beijing China - the official opening ceremony
13th July 2001 Moscow Russia - the date that the games were awarded to Beijing.
Timings of the 'birth' are more problematic. At what point during the day do we see the Olympics born, possibilities are sunrise, local noon (sun on the MC) or the Opening Ceremony, which has been set for 8:08 pm. Clearly the organisers believe that this is the birth time.
For the second date timings are again problematic. Do we take sunrise, noon, the time when Beijing won the second ballot or the time when the result was announced. Personally I think the time of the second ballot result is appropriate if that date is taken but can't find a reference to it.
These are not the only possible birth dates - there's the date the decision was taken to place a bid, the date that construction work began on the Olympic facilities, the time and date of the first event, etc.
There should be some Astrological effect so it's worth trying out some of these possibilities and then superimpose the eclipse positions to check for any result.
Edited to add:
If we take the start of the opening ceremony as the birth point, then Jupiter rules the Pisces Ascendant and is placed in the eleventh house of friendships. Jupiter also rules the Sagittarius MC. There is some indication that this should be a friendly games but Jupiter is not strong and is disposited by Saturn, which opposes the Ascendant and also rules the unfortunate 12 house. Saturn is also conjunct Venus, which rules the eighth house, also unfortunate as it relates to death.
Sun Mercury and Venus are in the unfortunate sixth house, with the Sun conjunct the unfortunate South Node.
The eclipse falls in the Sixth House, adding further emphasis. And seen from Beijing the Ascendant of the Eclipse chart is ruled by Saturn and conjoins Jupiter in the eleventh.
The prominence of Saturn, the planet of restriction suggests that these games will suffer from too much security and control. The unfortunate placing of the eclipse adds to this and the games may be in danger of self destruction (Saturn ruling the twelfth of self undoing) or at least making itself ill (Sixth House prominence).
Now this is only one possible birth time and date, so this reading should not be seen as definitive in any sense. However from a Western Astrological perspective the choice of start time for the opening ceremony is not good.