Some budget hosting companies offer exceptionally cheap hosting. Does it make any difference to my ranking who hosts my website?
In essence, no it doesn't matter who hosts your website, so long as your website can be easily and readily accessed.
However, if you use a low budget hosting company, you might be surprised to find out the location of the server hosting your website. It might not be in the country your hoping to attract traffic from? For example, if you have a plumbing and heating company in London, but your website is hosted in Dallas, Texas, it's unlikely that your really going to get the traffic you were ideally hoping for. So, you should always check the location of the server your website will be hosted on.
You may also want to consider how many other websites will be hosted on the same IP address (same server) your website is on? Some of the budget hosting companies make a profit by running staggeringly high numbers of websites on each server. Perhaps as many as 10,000 per server. Technically that might not be a problem if the hardware is capable of running all those sites, but can you be sure that your site is in a good neighbourhood? Typically, none of the sites on the servers will be monitored. If some of them are dubious in some way, or under an active penalty of some kind, your own website could end up suffering too.
When it comes to hosting, better to spend at least a reasonable amount of money with a company you know and trust. If it's a business website, you should get business class hosting and support. For a hobby or family website, it might not be so important.