It depends what you tell them. Santa Claus is an abbreviated form of Saint Nicholas. There are several threads already on this topic which you might read up on.

Personally, Santa Claus was never mentioned in our home. When my eldest was asked by an old woman what Santa Claus had brought him he just looked at her puzzled.

There is no harm in Santa Claus per se. It just depends on how you choose to treat gift giving and receiving.

Teach your children about St Nicholas whose feast I believe id December 6th.

We always taught our son that, no, there is not a literal "Santa Claus" who comes down the chimney, brings presents, and so on.  We taught him that St Nicholas does indeed exist, he is a saint in heaven, and can intercede for us, and that it is a custom to pretend that he brings presents (St Nicholas Day in Poland, Christmas Day in the US), and for men to dress up as him and play the role. 

It never created the slightest issue in our home.  I am not Polish but his mother is, and we incorporate customs from our respective cultures.  Polish children are not brought up to "believe in Santa Claus".

If you will stop and think about it, concocting a "Santa Claus" figure really smacks of Protestant misunderstanding of who saints are, and what they can do for us through their intercession.  Lacking this, they just pass on a lame story that, for some, becomes the first lie their children ever catch them up in.

Well, it's a lie, isn't it?

I haven't had kids yet and probably won't for a number of years yet, so this isn't especially pertinent for me, but I wonder what both Catholic parents and of course those theologically educated/trained have to say about this very common practice in our society.

It is lie. That should be the primary reason not to entertain this secular fantasy legend. The secondary reason is that does take away the point of Advent and Christmas. I would also refrain from even referring to St. Nicholas as "Santa Claus". This is just another attempt to secularize Christmas, robbing it of all Catholic identity.

Our children gain nothing when we lie to them and will only contribute to losing their trust.

I have never even had a girlfriend but I wouldn't do it. Since it would be lying. It would be maybe better to tell stories about St. Nicholas and present like a fictional story or maybe mix some true stuff such as him slapping Arius for being naughty.