UK visit visa: What documents prove genuine relationship with the visitor?

At the very least you will want to attach your bank statements. Even though you plan only to provide them with a room, it must be clear for the ECO that your situation in the UK is well and legitimate. Although it seems a bit weird to me that you are willing to sponsor the room but not willing to support them too. Is that something outside of your means to offer? I ask because of V 4.4 of immigration rules:

V 4.4 The third party may be asked to give an undertaking in writing to be responsible for the applicant’s maintenance and accommodation. In this case paragraph 35 of Part 1 of these Rules also applies to Visitors. An applicant will normally be refused where, having been requested to do so, the applicant fails to provide a valid written undertaking from a third party to be responsible for their maintenance and accommodation for the period of any visit.

So it's something you should be ready to offer.

As for proving ties, this is a bit more tricky. To find more details, I will reach for version 5.0 of Visit Guidance, namely the page 15 is what interests us here. There you can read:

To assess whether the relationship is genuine and whether the third party intends to provide support consider:

• the third party’s previous history of ‘sponsoring’ visitors - for example, previous failures to support visitors may call into question their intention and ability to do so for this application
• the relationship between the applicant and the third party

And continues with

For visa applications, you may want to check with the applicant or third party:

• where they met
• how often and by what method they communicate

So from the sound of it being able to show that you are close with your brother in law and friend, as you communicate often and have known one another for a substantial length of time is the key. I imagine that in establishing that a long history on Facebook chats, exchanged emails or record of phone calls together will help, but what they are really looking for is previous visits. Do you fly to visit your brother in law and your friend? Or do you travel somewhere together? Including a list of dates, places when you have travelled to visit them, or they have travelled to visit you before, will go a long way establishing that it is a genuine relationship.

Additionally I think that a solid proof can come from the relationship itself. As I was thinking about how would I prove my close ties with a friend I could start by telling the story how we met in an online game 13 years ago. I could then say how we talked outside of the game, met for beers and travelled together across Europe - seeing how many bars we can get kicked out from. How then in recent years I have flown across the globe just to attend his wedding. When you are close to someone, then there are plenty of memories that you share. And when immigration officer asks about the relationship will be the time to share them.

The reason for their visit to the UK is also crucial and it must be clear to the Immigration Officer that it is indeed an important occasion, as without it the application is likely to be refused.

When inviting your family member or a friend to the UK you do not necessarily have to provide maintenance and accommodation to the invitees (or mention in the invitation letter that you would).

1. Can write only a letter of invitation to support their application - this letter can contain only your invitation, reason for invitation, duration of the visit, relationship to the invitee and employment of the invitee (just to re-confirm) and may be a travel itinerary (time schedule what you plan to do when they visit).

2. Can agree to provide only accomadation. For this you do not need to provide proof of your financial status. But the invitees should be able to provide evidence that they have sufficient funds to support themselves while in the UK. Also it is better to mention in your invitation letter that your invitees will be using their funds for the trip just to be clear.

3. If you agree to provide maintenance and accommodation - you have to provide evidence of financial situation (bank statements or fixed deposits - no need of utility bills)

These are my findings for now. I did not find any specific answer for the proof of genuineness of the relationship.

Here are the guidelines for supporting documents.