As a rogue, I've gained enough XP to level up to level 3. However, I'm not yet ready to pick what archetype would fit best for me.
Despite having gained enough XP, can I wait until later to level up my character?
I'm looking for rules to back me up, especially if it explicitly states that I don't need to level up immediately upon acquiring the XP threshold.
The PHB states:
As your character goes on adventures and overcomes challenges, he or she gains experience, represented by experience points. A character who reaches a specified experience point total advances in capability. This advancement is called gaining a level.
Note that it says "a character [...] advances in capability", not "a character can advance in capability". Contrasted with many other features that are optional, this means (RAW) that you can't delay your level-up.
Furthermore, Xanathar's Guide to Everything has a variant rule for shared campaigns, so-called checkpoint leveling (see XGtE p. 173). It's basically another alternative to XP, much like milestone leveling, so I won't go into details; its description says the following, however (emphasis mine):
At the end of a play session, characters must level up if they have reached enough checkpoints to do so.
This strongly suggests that leveling up as soon as you can is not only RAW, but also RAI (rules as intended).
I haven't found anything else targeting your issue in either the PHB, DMG or XGtE.
Since the rest of your party probably doesn't want to wait, you'll be one level behind them. That's not a problem, it's quite common if the DM awards XP individually to everyone. However, you have to make sure to ask your DM if delaying your level-up is ok: it will affect their encounter balancing, especially if your DM normally makes sure that everyone levels up at the same time, for example by using milestone leveling.
For the same reason, mid-session-level-ups are also a no-go.
The DM may also have something planned that strongly requires you to level up. For instance, if you're level 4 and the big boss battle is coming up, you shouldn't be delaying your level-up, as level 4 to 5 is a pretty significant jump in power.
Therefore, if your DM says "no, you can't delay your level up", you should accept it without objecting—they likely have their reasons, not all of which they may be able to reveal to you.
This is a bit of an XY Problem.
The title of the question is 'can I defer leveling up?' but the actual problem is expressed in the second sentence:
Talk to your DM about your actual problem - you don't want to choose an archetype yet. Tell them that you are ready to assume all of the generic mechanical advantages that come with level 3 (hp, class features such as rages, ki points, or in your case, another sneak attack die) as well as the effects for them (treating you as level 3 for calculating encounter CR, etc.). Ask them if you can hold off on selecting your archetype and assuming the specific benefits that come with that. If not, ask them whether you can 'try out' an archetype for a while before making your final decision, and still have the option to switch to another one later.
Update: The release of Tasha's Cauldron of Everything means that there are now (optional) rules for changing your subclass. You are permitted to do this "when you would normally gain a new subclass feature". This is obviously not a good option for you, even if permitted, since rogues gain subclass features at 3rd and 9th levels. Thus if you were forced to make a subclass declaration at 3rd and were not satisfied with the result, you would need to wait until 9th to change your subclass according to this rule.
As PixelMaster's answer already addressed, under the default rules characters level up once they have enough XP. Under a checkpoint leveling system, they automatically level up after reaching enough checkpoints. However, there is a variant rule from the Dungeon Master's Guide that describes additional requirements to level up.
Under the list of additional downtime activity options that the DM can choose to make available to players, the "Training to Gain Levels" section says (DMG, p. 131):
As a variant rule, you can require characters to spend downtime training or studying before they gain the benefits of a new level. If you choose this option, once a character has earned enough experience points to attain a new level, he or she must train for a number of days before gaining any class features associated with the new level.
Under this variant rule, you could choose to not undertake the training right when you gain enough XP to level up. By doing this, you could defer leveling up for as long as you'd like. However, this would require the whole table to operate under this ruling. If you're playing in an already-established campaign, there's no way to choose not to level up. My advice to you would be to talk to your DM about putting the choice off until later - maybe see if they're alright if you test out the subclass you most like in the next session, and then change your mind if you don't like it.
As a compromise between the two answers provided so far (which have already provided the rule-based answer you seek…keeping in mind that of course, at the end of the day what matters is what you and your fellow gaming friends decide, especially the DM), consider making the non-optional changes to your character now, and defer the archetype selection until later. I.e. this doesn't have to be either/or.
It still makes sense to run this by your DM, because I agree that they might have some good reason not to allow it. But that would be very unusual, especially at level 3.
From an in-character point of view, this is easily justified by the fact that basic abilities tend to be acquired over time through general experience, while the selection of archetype is a conscious choice made "at the crossroads", i.e. at the point in your character's development where it makes sense to start specializing. You can't help but get better at fighting (like gaining another d6 on sneak attack) but other character changes might require more active thought (such as choosing an archetype or, at later levels, figuring out what ability score will improve through focused training).
By doing it this way, you defer the main thing that you want to have more time to consider, but your character will have some of the advancement benefit needed to keep your party strong enough to meet the challenges ahead.
As a DM I have allowed players who are not sure which archetype they want to change there mind until level 6. It might be worth talking to your DM to see if this is acceptable. I have a few caveats I don’t necessarily let them pick and choose each session but instead at appropriate moments in the adventure sometimes providing specific situations that allow them to experience a particular archetypes strengths and weaknesses helping them make a decision.
It might be worth Discussing a similar option with you DM as a solution.
