There's something called the
Network Railcard, which gives a third off most rail fares in the south east of England for the holder and up to three accompanying adults and four accompanying children, though only after 10am on weekdays (and furthermore, on weekdays there's a minimum fare of £13). It costs £28 and is valid for a year - it's supposed to only be used by the named holder (the coverage map is
here).
The Network Rail discount applies to fares on both
Heathrow Connect and
Heathrow Express - it would reduce the cost of an adult return on Connect from £17.80 to £13 on weekdays (or £11.75 if both the out and return journeys are made during the weekend or on a public holiday).
However buying a Network Railcard if you're arriving from outside Britain is likely to be an issue, as I'm almost certain that the Heathrow Express/Connect ticket offices at LHR don't sell them.
Railcards can be bought from any 'normal'
National Rail station ticket office, but alas the Heathrow ticket offices are an exception as they don't count as 'normal' in this regard. A few train companies do still have telesales lines which can I think sell Network Railcards, but I rather doubt they'll dispatch orders to non-UK addresses.
If you want to save a bit of money, I'd suggest just taking the Tube - it's really not that big an ordeal. (Non-Oyster fare is £5.30, Oyster fare is £4.80 or £2.90 depending on time of day - all Tube fares are singles.)