I need advice about obtaining a pay-as-you-go SIM card for use in my mobile phone while in Paris.  What would be the cheapest option?

Your phone needs to be unlocked from its current network

SIMs will be quite easy to find in plenty of shops, like here

There are 3 main networks and a number of virtual providers including supermarkets. You can compare them at http://www.prepaidgsm.net/en/france.html

As you'll see, the ordinary tariffs aren't that good, but if you look more deeply on the network's own sites you'll find some useful add-ons, e.g. Orange illimité inclusive packages, or SFR has an option with a 30 cent charge for up to an hour to landlines and SFR. This would be good with a local access number for a calling card or similar arrangement to call back to the UK.

Another way to make cheaper calls would be to use it with a callback service, which sets up 2 calls, the first to your phone, then to the destination

French SIMs tend to be short-lived if not regularly topped up, so your choice might also depend if this is for a one-off visit or at regular intervals

Another possibility, and especially if you'll visit other countries as well, is that  there are several global roaming SIM cards with free incoming calls in many countries, and outgoing rates from about 20p a minute. Some of these have Isle of Man or Jersey numbers, which come from some but not all UK network's inclusive minutes. I can add more about this  if you like

Or you could just use a UK SIM. 3 options worth looking at are: O2's My Europe Extra, with £10 a month fee then free incoming calls, outgoing 25p/min; Vodafone Passport, with 75p connect fee, then no further charge for incoming, same rates as at home for outgoing; or 3, with the cheapest ordinary roaming at 10p/min incoming, 25p/min outgoing

Another possibility, and especially if you'll visit other countries as well, is that  there are several global roaming SIM cards with free incoming calls in many countries, and outgoing rates from about 20p a minute. Some of these have Isle of Man or Jersey numbers, which come from some but not all UK network's inclusive minutes. I can add more about this  if you like

Yes please!  I found your suggestions very helpful.  We are planning to go to several destinations in Europe with friends who tend to 'get lost' with great rapidity and it would be most useful to keep in contact with them while abroad.
Thanks very much for your knowledgeable advice!