There was a lot of rain. Flooding / trains cancelled - biggest thunderstorm I've seen in the UK since moving here 13 years ago. In this case I think it probably made a difference, because those that were fighting to leave were more likely to turn out.
Arguments for would be learning something new, or breaking information during the period of the vote. For instance, if a news story broke about a politician, or something hidden in the wording of a bill would be highlighted, then one might want to change their vote.
Also, propaganda/information campaigns would ramp up towards the end of the vote, which could be good or bad.
Arguments against are basically logistical burden and opportunity for fraud. It's simpler for one vote, cast in stone.
However, assuming one cannot change their vote, I disagree that there is more opportunity for tampering. Each count of the vote is basically a statistical sampling, and it should stay relatively the same.
Here in the US, all elections are handled locally, at the county level. If there is fraud, it's hard to detect the day of the election, so the fraudulent vote becomes part of the official ballot count. Sure, months later, poll workers, voters, and politicians can be prosecuted for election fraud, but at that point, the election has already long been decided. It's very rare that a real recount is held that becomes the actual result. It's easy for fraud to become the accepted results.
However, with a long voting period with daily rolling counts, there is more opportunity for scrutiny, statistical methods, recounts, etc. If anything funny happens, it will be easier to detect with more time, and prevent it from becoming the official results.
That way, it's not just a high-stakes, one day event. Everyone will have a chance to vote, and nobody can (rightfully) say, "I wish I had voted, it didn't go the way I wanted" or "If I had known, I would have voted". You will know, and there will be time for you to vote still.
If I understand correctly, there were heavy rains that affected the turnout for this vote. Should something this important really be that affected by the weather?