The speed at which a ship or probe would need to travel to create a noteworthy amount of time dilation depends on the level of time dilation you consider "noteworthy" and the context in which it is being observed. However, I can provide you with some examples to give you an idea of the speeds involved.
In order to observe a significant time dilation effect compared to a stationary observer, the object in question needs to be traveling at a substantial fraction of the speed of light, which is approximately 299,792 kilometers per second. Here are a few examples:
For a time dilation effect that is noticeable but not extreme, such as a time dilation factor of 2 (meaning time passes twice as slowly for the moving object), the object would need to travel at speeds around 87% of the speed of light.
To achieve a more pronounced time dilation effect, such as a factor of 10 (time passing ten times slower for the moving object), the object would need to reach speeds of approximately 99.5% of the speed of light.
If you desire an extremely significant time dilation effect, such as a factor of 100 or more, the object would have to approach speeds very close to the speed of light itself.
It's important to note that as an object approaches the speed of light, the amount of energy required to accelerate it further increases exponentially. This means that reaching speeds close to or exceeding the speed of light becomes increasingly difficult in terms of energy requirements.
Additionally, keep in mind that the relativistic effects of time dilation become more pronounced as the speed of the object increases. However, the specific values and effects can be subjective based on the context and narrative requirements of your story.