Should I put number combinations like 1111111 onto my lottery ticket? Suppose the winning combination consists of $7$ digits, each digit randomly ranging from $0$ to $9$. So the probability of $1111111$, $3141592$ and $8174249$ are the same. But $1111111$ seems (to me) far less likely to be the lucky number than $8174249$. Is my intuition simply wrong or is it correct in some sense?
You should never bet on that kind of sequence. 
Now, every poster will agree that the odds of any sequence from 000000000 through 999999999 has an equal probability. And if the prize is the same for all winners, it's fine. But, for shared prizes, you will find that you just beat 10 million to 1 odds only to split the pot with dozens of people. 
To be clear, the odds are the same, no argument. But people's bets will not be 100% random. They will bet your number as well as a pattern of 2's or other single digits. They will bet 1234567. I can't comment whether pi's digits are a common pattern, but the bottom line is to avoid obvious patterns for shared prizes.

When numbers run 1-50 or so, the chance of shared prizes increases when all numbers are below 31, as many people bet dates and stick to 1-31. Not every bettor does this of course, but enough so shared prizes show a skew due to this effect. 

Again - odds are the same, but human nature skews the chance of split payout. I hope this answer is clear.
As mentioned by all other people here there is no disadvantage in picking 11111111 as your lottery number.
However I would just like to add that there is a pattern in these kinds of thinking mistakes.
Transitivity
Let's say that :

A = a very unlikely event
B = a very unlikely event

Our brain is tempted to use transitivity whenever it can.
Immediatly we want to know what happens if A and B would happen at the same time. Usually, the chance of A and B happening at the same time would then be even smaller. We would have to multiply the chances.
Cards, An example of transitivity
When taking a random card from a deck of cards (52 cards):

The chance to take an Ace = 1/13
The chance to take a Spade = 1/4
The chance to take the ace of spades = 1/13 * 1/4 = 1/52

Here it makes perfect sense.
The lottery problem , transitivity ?
It is tempting to apply this reasoning to the lottery problem.

A = the chance to guess the lottery number is very small
B = a number with all equal digits seems like something rare

And here again, our brain is tempted to use transitivity
The reasoning would then be that ...

the combined chance of (A and B) is smaller than the chance of A.

Unfortunately this is incorrect. You cannot apply transitivity here.
But why not ? I will illustrate with some more examples.
Dice problem , another transitivity failure
Here is an obvious example that makes the same mistake:

the chance to throw a 1 with a dice = 1/6
the chance to throw a 2 with a dice = 1/6
What is the chance to throw a 1 and a 2 at the same time with the same dice ? Is it 1/36 ?

It is of course impossible to throw a 1 and a 2 at the same time.
When can you apply transitivity ?
So, what is the difference between the dice and the cards?
The difference is that (as with the cards) when applying transitivity we have to make sure that the individual conditions that we are combining do not interfere with each other. If there are no such dependencies then you can blindly apply transitivity and multiply the individual chances.
Applying it !
There is 50% chance that I am a man, there is 50% chance that I am a woman. However, the chance that I am both is not just 25%. This is a perfect example of interfering conditions.
So, yes a digit with all equal numbers is something special (0.00001%), and yes guessing the lottery number is difficult (0.000001%). The problem is that you are trying to combine these 2 conditions, but the conditions are interfering.  So in conclusion, even while it seems that you should multiply the chances (0.0000001 * 0.00000001), this is in fact a mistake.
I hope this explanation gives you a better insight.