There are several options here. I'll divide them into the options directly in the book, and other options I have established for my players.
There are a few official means for the party to get their hands on a large sum of money in the early adventure.
There is a secret room in the Zhentarim Hideout in Chapter 1. It contains 4 paintings worth 75gp each and 15 silver bars worth 50gp each--for a sum total haul of 1,050gp.
In addition to that, there is...
A much smaller amount of treasure hidden in the Xanathar Guild Hideout, totaling 26gp 7sp
This gets us pretty close to the needed amount simply with found treasure. As for making up the difference...
Istrid Horn, a Zhent, is listed as being available to offer loans up to 2,500 gp with an interest rate of 10% per tenday (See the introduction)
Some of the Faction missions pay as well...
The 3rd level Emerald Enclave mission pays 100gp per character who participates
The 3rd level Harpers mission gives a spellbook with 2nd level spells (see below, may be worth money to Obaya)
The 2nd level Zhentarim mission pays 50gp per character who participates
The 3rd level Zhentarim mission pays 15pp (150gp) flat rate
If you plan to tie into Dungeon of the Mad Mage...
Obaya Uday is offering gold for spellbooks, based on the highest level spell in the book. For a book containing only 1st level spells, such as the one that can be found in the Xanathar Guild hideout, she offers 5pp (50gp). For one with 2nd level spells, like the one acquired from the 3rd level Harpers mission, she offers 25pp (250gp)
From here on, these are all additional options I came up with for my players, while confining it to characters who exist in the book (and a bit of older info about them, in one case).
As for other sources of loans...
Mirt, of the Harpers is known as Mirt the Moneylender. He could probably be called on to lend the party money...and if a member of the party is recruited to the Harpers, he might cut them a good deal. The Cassalanters are also called out as moneylenders and that could lead to some very interesting gameplay if they are also the villains. Finally, appendix B names another family that serves as a moneylender: the Irlingstars. Appendix B also states that Istrid's interest rates are in-line with other money lenders, so again...10% per tenday (which is steep...ouch)
Beyond that, there's the option of sponsorship and investment...
Mirt is extremely wealthy...if the party can make it appealing, he might be persuaded to invest in the restoration for a percent of the profits. If you offer to make the Manor a safehouse/operations house for the Harpers, he might just cover your missing margin as an investment in the well-being of the Harpers. Any other nobles that the party has gotten to know (such as Raenar Neverember) could be possible investors as well. And, of course, any other Faction could potentially sponsor the repair of the Manor as well, as long as it benefited them appropriately.
Finally, while these options were available to my players, I left it open for them to be creative. When it comes to acquiring money, players can be quite inventive. I nudged them along to try to make sure they found all the hidden loot in Chapter 1, and that put them really close to what they needed. So, I simply asked them where they were going to get the rest of the money.
One player started selling his services as a spellcaster for a bit, and the Bard went on tour around town.
The only assumption made through the rest of the adventure is that the PCs can be found at Trollskull Manor. This is the setting piece of the Fireball event...because someone was coming to the players at the Manor when they were attacked. There is no assumption that it has been repaired or that it is back in business.
So, players can choose to renovate it now...renovate it later...or just live in a broken down half-ruin indefinitely.