Etiquette question: is there such a thing as accepting an offer 'too early'?

Actually, it is simple courtesy to accept as soon as you know you will do so. This helps the institution deal with its obligations to you and others. It isn't a game you play for "advantage". You hope to develop a long term relationship with this institution. Treat them with respect and expect them to reciprocate.

Don't accept before you are ready, such as when other offers might come in, but don't delay for any perceived "strategic" reason.

Similarly, once you have accepted the offer, extend the other universities the courtesy of letting them know you will not be accepting theirs.

I would take your colleagues advice - but not for the reason he cited.

Waiting a few days allows the excitement to die down and stops you from making rash decisions. Perhaps there is some catch in the agreement that you glazed over because you were too interested in the good parts.

You've also mentioned that other offers are coming in - waiting allows you to consider all of them properly. There might be a gem among them that you notice too late (if at all). It would be unfortunate to accept this offer moments before a better one arrives in the mail.

So I recommend waiting at least a couple of days just to cool down and consider your future carefully. If you have already done this, then I see no reason not to go ahead. As other answers have said, you want to build a positive relationship with these people and being prompt is a good way to start.

I doubt they will care much either way if you accept early or at the last minute. Appearing eager is not a bad thing. You should be excited to start a PhD.

It is courteous and prudent not to delay if your decision has really already been made. It means they and you can move forward with the process of registering as a student. It also means they can say no to waitlisted candidates.

However, if your decision is not certain yet, then don’t make it earlier than necessary. For example, you haven’t seen all the other options yet. And you haven’t visited. These could change your mind. So I would wait until you have a clearer picture.