In the past, I was never sore longer than 12 hours after intense workouts. However, yesterday I had something on my mind and as a distraction, I did:

2 hours of basketball
2 hours of biking
1 hour of weight lifting

As a result, I'm now pretty sore all over my body. Do you think it would be wise to skip today's working out and let my body recover for tomorrow? Or perhaps just some easy, low-impact exercycling to get my heart rate up?

I usually work through soreness with some easy cardio work. Once you get the muscles warmed up and ready to go, you will usually feel a lot better. But if you lifted weights yesterday, I defintely would not do any weights today.

i usually get through soreness with a light run-through of whatever is sore -e.g. legs - either elliptical or light weight squats - arms - if I'm sore from BP then I'll do light bench press or even DB bench , same for back, etc.
anecdotally this seems to help -

For basketball related leg soreness (jumping, running) biking could be good - but because you had enough biking yesterday I would go for a walk. (Min 30 min, but 60 min better.) For biking related leg soreness jogging could be good, - but again, you already had enough running yesterday, so I would recommend the same walk:-))
Without walk (or some other type of moderate leg exercise) this soreness might stay longer.

for me - just deep knee bends - about 10-15 markedly reduces soreness when quads & hamstrings are sore - assuming I can do them without hanging onto a door or a desk (when really really sore )

rub some muscle cream on your aches and move on and work at the same intensity as always...but hey, my advice could be bad for your health

Well I don't know if this is any help to you since it seems you are not really into muscle building exercise like weights, but since doing BFL I have learned the importance of letting muscles really rest and repair between doing the same workout - I suppose the difference is whether you are out to build muscle or burn calories - if you want to build muscle then you must let them repair properly before working them again, but I suppose to just burn calories working them before they have repaired would still work.

I work through it. No pain no gain! ha ha.
Drink lots of water!!!
jag

I decided against working through it.

My goal is to put on muscle at the moment. Boxing and basketball kind of counter that, but I won't be doing those 2 as often as I do weight lift.

Today I feel great again and will return to my weight routine.

Glad you feel better. I think recovery days are actually beneficial when I overdo it. If I feel too stiff, I put my bike on the trainer and ride really slowly in an easy gear just to get moving a little, or I walk at a real slow pace around the neighborhood or on the treadmill.