Both Eddymage's and matszwecja's answers offer worthwhile suggestions for what to do with the spellbook you mention, including selling it, using it as a plot hook, and several other possibilities.
In addition, I note that you are running Waterdeep: Dragon Heist, based on both your description of the spellbook:
I'm running a prewritten module, and after a first level boss fight the party found a spellbook with several spells in it (burning hands, disguise self, false life, shield, unseen servant, and witch bolt).
which exactly matches the spellbook of Grum'shar, a low-end criminal in Chapter 1, "A Friend in Need", Q7, Boss Fight; and also because you said as much in comments to Eddymage's answer.
There is information in Waterdeep: Dragon Heist, where this spellbook is from, which is useful in giving the party options for selling or otherwise disposing of it
This information will hopefully prove useful to you or anyone else running Dragon Heist.
Valuing the Spellbook
As noted in Eddymage's answer, in Waterdeep: Dungeons of the Mad Mage, Obaya Uday will buy spellbooks, and according to that adventure, Obaya will pay based on the highest level spell in a spellbook. So according to that metric, this spellbook is worth 50 gp.
However, there is a better metric available in Dragon Heist.
Using the Book Wyrm's Treasure to value the spellbook
The Book Wyrm's Treasure in Trollskull Alley, described in "Chapter 2: Trollskull Ally", is a bookstore in an alley familiar to the PCs.
Rishaal, the proprietor, makes spells for wizards to copy available for a fee:
Rishaal has a small collection of spellbooks and allows wizards to copy spells from them at the cost listed in the Spells for Sale table.
Rishaal allows wizards to copy first level spells for 25 gp each, and this spellbook has 6 first level spells in it. Using that as a metric, this spellbook has a nominal value of 150 gp.
Of course, nominal value is one thing, but in the end, something is only worth what someone is willing to pay for it.
Selling the Spellbook
There are multiple people in Waterdeep to whom the PCs should be able to sell that spellbook.
Selling to Rishaal, at the Book Wyrm's Treasure
Rishaal is an obvious possibility, and maybe the most likely possibility. He might be willing to buy the spellbook and add it to his small spellbook collection. He doesn't have four of the spells that are in this spellbook, so acquiring the spellbook from the PCs would allow him to expand his spell offerings.
How much is Rishaal willing to pay? We have no idea. We can speculate -- for instance, if you assume Rishall can sell access to a given first level spell once a month, then such a spell would generate 300 gp a year. But if he can sell them much more often, say, once a tenday, then a first level spell would generate 900 gp a year. But any speculation is just that. However, it's certainly reasonable that Rishaal would be interested in buying. After all, his collection of spells is very small, and growing it can only make his business more profitable.
So how to set a price? You can let your story be your guide. If you want the PCs to have a bit less money, maybe Rishaal is willing to only pay 100 gp or 200 gp. If you want them to have a bit more money, maybe he's willing to pay 500 or 1000 gp, or even more. Maybe it's worth that much to expand his collection, or maybe he doesn't want someone else to set up a competing business, so he's willing to pay premium.
So, somewhere in the neighborhood of 500 to 1000 gp is reasonable, enough to be a windfall for a party of first level characters, but not an incredible amount of money. But really, you have a great deal of latitude here.
Regardless, selling to Rishaal is an excellent option, likely to make Trollskull Alley more interesting and vibrant to the PCs, and may well give them a valuable connection for selling future spellbooks, or for anything else.
Selling to Obaya Uday
We've already established that Obaya is willing to pay 50 gp. And she's right there in the Yawning Portal. But only 50 gp! What a cheapskate! But we know she's not a practitioner herself, and has little interest in the actual contents of the spellbook, so her value is really a floor.
Selling to Obaya should be easy. The players may already know Obaya from the Dragon Heist Introduction, section "Familiar Faces", where it says:
Give the Yawning Portal Familiar Faces handout in appendix C to the players, and allow each of them to select one NPC as a friendly acquaintance — someone the player’s character knows and trusts. More than one player can select the same NPC.
Even if none of the PCs selected Obaya, any of the other "familiar faces" NPCs (Or Volo, for that matter) might easily point them in her direction.
Unless you have a particular plot reason for the PCs to sell to Obaya, or the PCs do, selling to Rishaal is likely a better option, both monetarily, and for story advancement.
Selling to someone else
The PCs might be able to find someone else willing to buy. The PCs have many opportunities to establish connections in Waterdeep, who might point them to someone willing to sell. In addition to the "familiar faces" at the Yawning Portal, and to anyone they might happen to meet around town, there are ample opportunities to the PCs to have other connections.
Some of the PCs may have guild or faction membership. See the section on guild membership or the https://www.dndbeyond.com/sources/wdh/introduction#FactionMembership in the Introduction.
Finally, both Volo and Renaer have connections that might help the PCs find someone to sell the spellbook to, as described at the end of Chapter 1, in "Fine Friends".
What would someone else pay? Hard to say, but probably not much less than what Obaya will pay, and probably not much more than what Rishaal would pay.
The price
Using Obaya and Rishaal to set soft lower and upper ranges for what the PCs might sell the book for gives us the very wide range of 50 gp to 1000 gp. Someone might offer less than 50 gp, if they think the PCs will sell, because the buyer could turn around and sell it themselves to Obaya. Someone might buy it for more, if they have specific reason to, but the very speculative upper range I gave for Rishaal of 1000 gp is probably a good upper bound.
Otherwise disposing of it
Maybe, just maybe, the PCs find something more useful to do with the spellbook than sell it. Maybe they can use it for leverage with some NPC. If so, that would be a fun and useful further development of the story.
Or they could sell it.
What to do with the money
So, let's say you've given the PCs 1000 gp, or 2000 gp. What will they spend it on? There are lots of options. Trollskull Manor is a hole to throw money into, for sure. The choices are endless. After all, Waterdeep is the City of Splendors, the Crown of the North, and the PCs are likely to come up with something fun to spend the money on. After all:
Almost anything can be bought or sold in Waterdeep. There’s no need for adventurers to shop elsewhere.
