No, spells must be prepared when you finish a log rest.

The section on Preparing and Casting Spells for the wizard states:

you can change your list of prepared spells when you finish a long rest.

If it said something like "anytime after you finish a long rest" then sure. But as it's worded you can only change your list of prepared spells when you finish a long rest. Failure to do so leaves you with the same set of spells you had prepared the day before, not open slots to fill later.

This is also backed up by Jeremy Crawford, lead rules designer in this twitter exchange:

Lebedev Dmitry
^@LebMit
@JeremyECrawford Can spellcasters prepare spells not all at once, but leave some "slots" for preparing spells later during the day?

Jeremy Crawford
^{‎@JeremyECrawford}
You prepare your list of spells only at the end of a long rest.

This is one of those changes from 3.5e to 5e D&D

There was a mechanic in D&D 3.5e wherein a Wizard or other spellcaster that uses the spell preparation mechanic could in essence leave a few of their spell slots unprepared so that later they could be filled as necessary with spells that they might need throughout the day. These "late preparations" took a minimum of 15 minutes but allowed the spellcaster to get a deal of utility out of spells that had a non-combat role. The caveat was that the slots had to be empty and you couldn't drop a slot you had prepared a spell in to prepare a new spell unless you had just finished a full rest period.

The new edition introduced ritual spells to cover some of the need for this

Preparation in 5th edition changed a great deal from prior editions of Dungeons and Dragons. The 15 minute spell prep by leaving spell slots empty isn't really required anymore due to the ritual mechanic which allows using 10 minutes to cast non-combat spells (that have the ritual tag) outside of combat. This provides the Wizard with a lot more versatility on how to choose the spells they'll use to deal with the challenges they believe they'll face that day. (Gonna be in the middle of the desert? Probably won't need feather fall.)

The 5e rules also opened up your spell list so that you can prepare less of one spell level and more of another, giving many choices on how many spells of each level you'd like to prepare. You are no longer locked into 4 first level spells and 2 second level spells. You could choose to prepare 2 first level spells and 4 second level spells so you have more of the useful spells available to account for your daily challenges and to get the most out of your higher level spell castings.

What is your DM's preference?

Fifth edition never states that you can't prepare five of your six daily spells and leave the six slot open to prepare it later as you may need it. It also doesn't explicitly say that you can do that either. It's a decision that will most certainly have to be made by your GM. For instances where the book doesn't explicitly state what you can and can't do, the GM is responsible for making a call that might or might not allow you to do that thing.

Is it needed?

The usefulness of this is actually a great deal less than in 3.5. Due to the availability of ritual spells and the greater flexibility in spell preparation, there is less of a need for leaving empty slots in 5e than there ever has been in D&D. In short, the introduction of ritual spellcasting has removed the need for an empty slot / late preparation requirement that just wasn't present in prior editions of Dungeons and Dragons.

Each caster class is treated a bit differently.

Check the PHB for specifics for each class to understand how the ritual tag applies.

Ritual Casting (Wizard) Basic Rules p. 30
You can cast a wizard spell as a ritual if that spell has the ritual tag and you have the spell in your spellbook. You don’t need to have the spell prepared.

The Cleric doesn't get the same benefit, for example.

Ritual Casting(Cleric) Basic Rules p. 22 You can cast a cleric spell as a ritual if that spell has the ritual tag and you have the spell prepared.