In the event of death: tips on what to say, and why

The simplest deep condolences are often the best:

"I'm terribly sorry
for your loss."

There's no need to verbally follow this up with vague offers of 'anything I can do'. In too many cases, this slides by unattended.

Instead, offer a sincere, quick, no–response–necessary statement such as:

• "Contact me, if any need arises."
• "Chloe and I were great friends once."
• "Jim was a pleasure to know."

Each of these examples (or something close that you devise) carries a unique tone, one of which will be most appropriate for your relationship with the deceased. 

To be specific offers more comfort. 

"Contact me" leaves the method of contact open to they who need relief—whether to call, write, perhaps visit.

Acknowledge that needs do arise, while removing the implication of need on the part of the grieving, as they likely would reject any appearance of neediness. The idea, at this point, is to be as little intrusive as possible. Be a quiet shoulder, adding no burden.

Ambiguous passivity, awkwardness, and some predictability exist in offering "anything, if". You want to offer something, including yourself.

Authenticity is in the Follow Up

As with most any matter in effective communication, authenticity is found in caring. In due time, which could be a day, a week, a month (again, the relationship matters) check in again.

You'll have a chance in the meantime to gauge the immediate circumstances of the person(s) in mourning, and to offer specific help or company . . . whether to show up with a covered meal and a deck of cards (prepared to play or leave) or to meet up for a movie, maybe coffee. You get the idea. This is an effort to establish that you do continue to think about them and care that they're coping with their loss. 

On the other hand, when we're not an immediate family or friend, coworker perhaps, who will be present at some point during most days after a death—if it'd be a month, or three, before expecting to see them again—then it's okay to pick a time in-between for a reach-out. Too, if these were more distant acquaintances, then usually a personalized card will suffice.

Most people understand how difficult it can be to extend ourselves at all in a world where so much is misunderstood. All in all, just reach out. Go on record as the caring person that you are, and let everything else happen as it may.

"Abstract Flowers" image by Ken E, via PublicDomainPictures