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I got an invite from our client in the US to visit them for a "week" as per invite letter. This is on a business visa. My brother is there in the US and he is around 800 miles from the place where I am going to. In the invite letter, it is mentioned that I am being invited for a week at their premises for discussions. I am supposed to return on the next day (Saturday) flight. If I extend this to Sunday, I would be able to spend a day with my brother and his family, and can come back to the same airport to board a plain back to India.

Can someone advise on the legalities of this? I know that on a business visa, one can stay for 45 days in US, but the client invite letter says that it is for a week. Please advise. Also is it okay if I spend some personal time around to visit my relatives? Sorry if it sounds silly.

Since about half of the information is missing from the other answer and it is completely unsourced, let's gather it, the comments and sources into an answer:

tl;dr: irregardless of your visa, if you are admitted at all then you most likely will be admitted for six months but at least one month and you can do tourist-y things.

All my claims come from .gov domains.

Here's the official Visitor Visa page.

Visitor visas are nonimmigrant visas for persons who want to enter the United States temporarily for business (visa category B-1), tourism, pleasure or visiting (visa category B-2), or a combination of both purposes (B-1/B-2).

further down on this page you can find:

A visa allows a foreign citizen to travel to a U.S. port-of-entry (generally an airport) and request permission to enter the United States. [...] If you are allowed to enter the United States, the CBP official will provide an admission stamp or a paper Form I-94, Arrival/Departure Record. [...] You must depart the United States on or before the date indicated on your admission stamp or paper Form I-94, unless your request to extend your stay is approved by USCIS.

Note: I-94 has been automated for "air and sea" so unless you cross over land into the USA you will not get a paper I-94, consequently the admission stamp will tell you how long can you stay. There is an Electronic I-94 site but as per the above paragraph, the admission stamp is what matters. (For example, the electronic I-94 record might be missing.)

One thing a sole B-1 visa would differ from a sole B-2 or combined B-1/B-2 is the minimum length of stay granted. The USCIS page on B-1 Temporary Business Visitor has:

Initial Period of Stay: 1 to 6 months; 6 months is the maximum

And 8 CFR 214.2(b) has:

Any B-2 visitor who is found otherwise admissible and is issued a Form I-94 will be admitted for a minimum period of six months, regardless of whether less time is requested.

Finally, whether you can do tourist-y things, the answer is a yes but it's really hard to find an official source to it and it's stated in a truly bass ackwards manner. But here's what the USCIS says on Change My Nonimmigrant Status:

You do not need to apply to change your nonimmigrant status if you were admitted into the United States for business reasons (B-1 visa category ) and you wish to remain in the United States for pleasure before your authorized stay expires.

While it is not pertinent to this question, it is worth mentioning that it does not say you can freely do all you can do with a B-1 visa if you were admitted on a B-2.

You can stay in the USA for the duration you will get at the airport which will be on your I-94, typically six months.

It is perfectly normal and usual to combine business with pleasure in the USA. Don't worry about the dates on the client invitation letter. It is not the consular who will determine the length of your stay, it is the immigration officer at the airport. You can be issued hypothetically a one day visa by the consular but given six months stay once you reach the airport in the USA. The US visa issued by the consular only gives you permission to appear at a US border to request to be let in. Everything else depends on the border immigration officer.

Just answer the immigration officer honestly and he will likely give you six months IF the visa is issued.