How much approx would it be to take a taxi from Heathrow airport to
see the Eye (ferriswheel) in London? (aprox date July 2006)  Is there
any form of cheaper transport other than bus and or tube? Tours?

Try 
http://www.dialacab.co.uk/default.asp?page=LBC
Black cabs cost between 50 and 60 pounds
Other licensed mini cabs cost 25 to 35 pounds
The most comfortable alternative though I suggest try 
http://www.heathrowexpress.com/ 
14 pounds to paddington station
then take a cab to the london eye another 10pounds

As a Londoner living just round the corner from Paddington, I agree
with garby's advice above, except for the last bit. There's always a
long, long queue for taxis at Paddington, and the roads through town
are very congested. It would be far quicker and cheaper from
Paddington to take the Bakerloo tube line direct to Waterloo, from
which it is a relatively short walk to the London Eye. One thing that
puzzles me, however, is why you wish to travel there direct from
Heathrow. Will you be on a stopover between flights; if so, you may
have a problem with luggage, as I'm not sure there's any left luggage
facility at the airport.

The Londoners may correct me, but I believe the Heathrow express train
will be faster and cheaper.  Something to consider in July is that
there may be a long queue of tourists at the "Eye" - which could be a
problem if you are considering riding it between flights.

it will take about 45 to 50 British pounds each way.  Alternatively,
you could take the Heathrow express which is 25 pounds for a return
trip.  Pick it up at Heathrow and you will take it to Paddington
station.  It will be much faster than a taxi if you are arriving at a
busy time.  At Paddington, you can either take a taxi ( 25 pounds
round trip ) ot take the tube ( underground train ) from the
Paddington station to the Westminster Station.  Take the Circle Line (
the yellow line ) from Paddington Station.  It will be about 5 pounds
round trip.  When you get to Westimster and exit the station, you will
be at the base of Big Ben and close to West Minster Abbey, Parliment
etc.  To get to the London Eye, turn left, cross over the Bridge and
the Eye will be on your left hand side.