As others have said, the cutting of hands seems to be a fairly common (and logical) thing to do in a Lightsaber combat (the hands being the forwardest part of the body).
This is confirmed by Lucas himself:
VANITY FAIR: What is the meaning of all the hands, arms, legs and heads being chopped off in the films that have emerged from Lucas' brain?
GEORGE LUCAS: That's what happens when you play with swords
In addition, it's not just Skywalkers who lose hands.
From this absolutely excellent compilation:
:
Table 1
Total count of limb-loss championship in the movie (only listing those with combined win+loss >=2 - see the second table below for all limb losses classified by loosee type):
Duelist | Limbs cut | Limbs Lost
--------------------------------------------------
Obi-Wan Kenobi | 9 !!! | 0
Anakin/Vader | 3 | 5 (incl. 2 legs)
Dooku | 1 | 2
Luke | 2 | 1
Table 2:
All Lightsaber limb losses classified by loosee type
Duelist | Limbs Lost (a+l) | Details
--------------------------------------------------------------------------
Darth Maul | 0 or 2? | He's cut in half. Some people count that as 2 legs.
Mace Windu | 1 | I'm sure EU has him cutting some himself :)
Luke | 1 |
Dooku | 2 |
Anakin | 3+1+1 and 2 | - Loses a real arm in Ep.II to Dooku
| - Loses 2 prosthetic arms in Ep.III and Ep.VI
| - Also, in "Dark Force Rising" Emperor cut
| off Vader's hand as punishment for Ep4 fail.
| - In Splinter of Mind's Eye Vader loses a hand
Grievous | 4 + 2 | 2 arms to Obi-Wan in cyborg form;
| 2a+2l in crash before becoming cyborg
Ponda Baba | 1 | Definitely not a Jedi - in a bar (cantina)
Zam Wesell | 1 | Definitely not a Jedi - in a bar
Stormtrooper | 1 | Yoda chops off one hand in Ep.III
Wampa | 1 | Definitely not a Jedi
Acklay | 2 | 4-legged monsted on Geonosis vs. Obi-Wan
Please note that many of the limbs lost in the table above were NOT in Skywalker family, or even among Jedi. I grouped them in the table this way (Jedi group, Non-Jedi-Lightsaber-Fighter group, Sentient Humanoid non-Jedi group, Animal group - though Wampas were possibly semi-sentient).
And I'm not even counting limbs lost NOT to Lightsaber combat (e.g. C3PO)
Mind you, that doesn't stop people from reading into it. Three most popular interpretations I've seen were:
Freudian analysis from post-modernist movie critics who said it was an allusion to castration and some problems Luke and Anakin had with their mothers.
The theory that having a cauterizing weapon with no blood shown helps with movies' MPAA ratings.
Some minor theories about significance of Vader/Luke hands (Vader's, as a symbol of him turning into a machine, and Luke's as a symbol of a threat of turning into Vader). Then somehow everyone involved just kinda ran away with the meme, but then they got an arrow in the knee.
However, there's no canonical support for any of these.