Are incomplete sprint items estimated, or split.
Are incomplete sprint card re-estimated so that only the remaining work is estimated on the card? So as an example, if it was 5 and there are 2 points remining to be complete, it is now a 2 and the 3 completed point are simply lost. This does sound ok from a planning perspective but no so good from a knowing your velocity perspective.
The other option I have heard is to split the card in to a card representing the completed work that stays in the current sprint and another that represents the remaining work that is then added to the back log.
Velocity is the rate at which work is Done, not partially Done. That makes it a useful measure. If an item is not Done, it makes no sense to split it unless the parts can be Done. Attempting to do so would make the measure less useful.
It's important to avoid becoming a story point accountant, and to focus on value delivery. The Product Backlog should tell the truth at all times about how much work is currently thought to remain for the Product. That's it.
You have just experienced the fallacy of using story points as an indicator of velocity. What exactly does 5 story points mean? It means that based upon the information known at the time the estimate was made, the Developers felt it was more work than a 3 and less work than an 8. Beyond that, it is meaningless. You also need to remember that once work begins, new information can be uncovered and the original estimate is no longer useful. That is probably what is happening in your situation.
As @Ian stated, velocity is the rate at which work is Done. Doing work uncovers new information that can cause plans to be changed (this is why the Daily Scrum is part of the Scrum framework). I have much more success using trailing indicators to determine velocity. Throughput and cycle time have been much more effective for me and the teams I work with. These books by Daniel S. Vacanti are very helpful in understanding and have helped many people understand. Start with "When will it be done".
The Scrum Guide has this statement:
If a Product Backlog item does not meet the Definition of Done, it cannot be released or even presented at the Sprint Review. Instead, it returns to the Product Backlog for future consideration.
If something is not completed, return it to the Product Backlog and handle it like anything else that is there. Refine it further and consider it for a future Sprint.
It's important to avoid becoming a story point accountant, and to focus on value delivery
Agree with the above statement made by @Ian. Dont attach too much importance to story points (which is an estimate anyway).
Even when it comes to the question of splitting you should check if the part delivered adds incremental value. Do not get into the trap where stories are split so that teams can take credit for the work being done by splitting and closing one part of the story (though no value is delivered)..the question to be asked is it really done ? is value being delivered ?
Reestimating comes into the picture when you have decided it is not wise to split the story as no value is delivered. Reducing the story points only for partially done items will help you during the planning of the next sprint.